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t´ınh dS.. Chuyˆ e’n sang to.a dˆo.. Xem v´ı du.. khˆong tuyˆ e.t dˆo´i.. khi v`a chı’ khi t´ıch phˆan suy rˆo.ng.. nhu.ng chuˆo˜i phˆan k`y.. nhˆa´t cu’a chuˆo˜i du.. ’ du.ng dˆa´u hiˆe[r]

B `AI T ˆA P

TO ´AN CAO C ˆA´P

Tˆa.p 3

Ph´ep t´ınh t´ıch phˆan L´y thuyˆe´t chuˆo˜i. Phu.o.ng tr`ınh vi phˆan

10 T´ıch phˆan bˆa´t di.nh 4 10.1 C´ac phu.o.ng ph´ap t´ınh t´ıch phˆan 10.1.1 Nguyˆen h`am v`a t´ıch phˆan bˆa´t di.nh 10.1.2 Phu.o.ng ph´ap dˆo’i biˆe´n 12 10.1.3 Phu.o.ng ph´ap t´ıch phˆan t`u.ng phˆ` n a 21 10.2 C´ac l´o.p h`am kha’ t´ıch l´o.p c´ac h`am so cˆa´p 30 10.2.1 T´ıch phˆan c´ac h`am h˜u.u ty’ 30 10.2.2 T´ıch phˆan mˆo.t sˆo´ h`am vˆo ty’ do.n gia’n 37 10.2.3 T´ıch phˆan c´ac h`am lu.o ng gi´ac 48

11 T´ıch phˆan x´ac di.nh Riemann 57

12 T´ıch phˆan h`am nhiˆ`u biˆe e´n 117

12.1 T´ıch phˆan 2-l´o.p 118

12.1.1 Tru.`o.ng ho p miˆe`n ch˜u nhˆa.t 118

12.1.2 Tru.`o.ng ho p miˆe`n cong 118

12.1.3 Mˆo.t v`ai ´u.ng du.ng h`ınh ho.c 121

12.2 T´ıch phˆan 3-l´o.p 133

12.2.1 Tru.`o.ng ho p miˆe`n h`ınh hˆo.p 133

12.2.2 Tru.`o.ng ho p miˆe`n cong 134

12.2.3 136

12.2.4 Nhˆa.n x´et chung 136

12.3 T´ıch phˆan d u.`o.ng 144

12.3.1 C´ac di.nh ngh˜ıa co ba’n 144

12.3.2 T´ınh t´ıch phˆan du.`o.ng 146

12.4 T´ıch phˆan m˘a.t 158

12.4.1 C´ac di.nh ngh˜ıa co ba’n 158

12.4.2 Phu.o.ng ph´ap t´ınh t´ıch phˆan m˘a.t 160

12.4.3 Cˆong th´u.c Gauss-Ostrogradski 162

12.4.4 Cˆong th´u.c Stokes 162

13 L´y thuyˆe´t chuˆo˜i 177 13.1 Chuˆo˜i sˆo´ du.o.ng 178

13.1.1 C´ac di.nh ngh˜ıa co ba’n 178

13.1.2 Chuˆo˜i sˆo´ du.o.ng 179

13.2 Chuˆo˜i hˆo.i tu tuyˆe.t dˆo´i v`a hˆo.i tu khˆong tuyˆe.t dˆo´i 191

13.2.1 C´ac di.nh ngh˜ıa co ba’n 191

13.2.2 Chuˆo˜i dan dˆa´u v`a dˆa´u hiˆe.u Leibnitz 192

13.3 Chuˆo˜i l˜uy th`u.a 199

13.3.1 C´ac di.nh ngh˜ıa co ba’n 199

13.3.2 D- iˆe`u kiˆe.n khai triˆe’n v`a phu.o.ng ph´ap khai triˆe’n 201 13.4 Chuˆo˜i Fourier 211

13.4.2 Dˆa´u hiˆe.u du’ vˆe` su hˆo.i tu cu’a chuˆo˜i Fourier 212

14 Phu.o.ng tr`ınh vi phˆan 224 14.1 Phu.o.ng tr`ınh vi phˆan cˆa´p 225

14.1.1 Phu.o.ng tr`ınh t´ach biˆe´n 226

14.1.2 Phu.o.ng tr`ınh d ˘a’ng cˆa´p 231

14.1.3 Phu.o.ng tr`ınh tuyˆe´n t´ınh 237

14.1.4 Phu.o.ng tr`ınh Bernoulli 244

14.1.5 Phu.o.ng tr`ınh vi phˆan to`an phˆ` n 247a 14.1.6 Phu.o.ng tr`ınh Lagrange v`a phu.o.ng tr`ınh Clairaut255 14.2 Phu.o.ng tr`ınh vi phˆan cˆa´p cao 259

14.2.1 C´ac phu.o.ng tr`ınh cho ph´ep thˆa´p cˆa´p 260

14.2.2 Phu.o.ng tr`ınh vi phˆan tuyˆe´n t´ınh cˆa´p v´o.i hˆe. sˆo´ h˘a`ng 264

14.2.3 Phu.o.ng tr`ınh vi phˆan tuyˆe´n t´ınh thuˆ` n nhˆa´ta cˆa´p nnn (ptvptn cˆa´pnnn) v´o.i hˆe sˆo´ h˘a`ng 273

14.3 Hˆe phu.o.ng tr`ınh vi phˆan tuyˆe´n t´ınh cˆa´p v´o.i hˆe sˆo´ h˘a`ng290 15 Kh´ai niˆe.m vˆe` phu.o.ng tr`ınh vi phˆan da.o h`am riˆeng 304 15.1 Phu.o.ng tr`ınh vi phˆan cˆa´p tuyˆe´n t´ınh dˆo´i v´o.i c´ac da.o h`am riˆeng 306

15.2 Gia’i phu.o.ng tr`ınh d a.o h`am riˆeng cˆa´p d o.n gia’n nhˆa´t 310 15.3 C´ac phu.o.ng tr`ınh vˆa.t l´y to´an co ba’n 313

15.3.1 Phu.o.ng tr`ınh truyˆ`n s´ong 314e 15.3.2 Phu.o.ng tr`ınh truyˆ`n nhiˆe.t 317e 15.3.3 Phu.o.ng tr`ınh Laplace 320

T´ıch phˆan bˆa´t di.nh

10.1 C´ac phu.o.ng ph´ap t´ınh t´ıch phˆan 4

10.1.1 Nguyˆen h`am v`a t´ıch phˆan bˆa´t di.nh 10.1.2 Phu.o.ng ph´ap dˆo’i biˆe´n 12 10.1.3 Phu.o.ng ph´ap t´ıch phˆan t`u.ng phˆ` n 21a

10.2 C´ac l´o.p h`am kha’ t´ıch l´o.p c´ac h`am so cˆa´p 30

10.2.1 T´ıch phˆan c´ac h`am h˜u.u ty’ 30 10.2.2 T´ıch phˆan mˆo.t sˆo´ h`am vˆo ty’ do.n gia’n 37 10.2.3 T´ıch phˆan c´ac h`am lu.o ng gi´ac 48

10.1 C´ac phu.o.ng ph´ap t´ınh t´ıch phˆan

10.1.1 Nguyˆen h`am v`a t´ıch phˆan bˆa´t di.nh

ta.i mˆo˜i diˆe’m cu’a khoa’ng v`a F0(x) =f(x)

D- i.nh l´y 10.1.1. (vˆ` su tˆoe ` n ta.i nguyˆen h`am) Mo i h`am liˆen tu c trˆen doa n [a, b] dˆ`u c´e o nguyˆen h`am trˆen khoa’ng (a, b).

D- i.nh l´y 10.1.2. C´ac nguyˆen h`am bˆa´t k`y cu’a c`ung mˆo t h`am l`a chı’ kh´ac bo.’ i mˆo t h˘a`ng sˆo´ cˆo ng.

Kh´ac v´o.i da.o h`am, nguyˆen h`am cu’a h`am so cˆa´p khˆong pha’i bao gi`o c˜ung l`a h`am so cˆa´p Ch˘a’ng ha.n, nguyˆen h`am cu’a c´ac h`am e−x2, cos(x2), sin(x2),

lnx, cosx

x , sinx

x , l`a nh˜u.ng h`am khˆong so cˆa´p D- i.nh ngh˜ıa 10.1.2. Tˆa.p ho p mo.i nguyˆen h`am cu’a h`am f(x) trˆen khoa’ng (a, b) du.o c go.i l`a t´ıch phˆan bˆa´t di.nh cu’a h`amf(x) trˆen khoa’ng (a, b) v`a du.o..c k´y hiˆe.u l`a

Z

f(x)dx.

Nˆe´uF(x) l`a mˆo.t c´ac nguyˆen h`am cu’a h`amf(x) trˆen khoa’ng (a, b) th`ı theo di.nh l´y 10.1.2

Z

f(x)dx=F(x) +C, C ∈R

trong d´oC l`a h˘a`ng sˆo´ t`uy ´y v`a d˘a’ng th´u.c cˆ` n hiˆe’u l`a d˘a’ng th´a u.c gi˜u.a hai tˆa.p ho p.

C´ac t´ınh chˆa´t co ba’n cu’a t´ıch phˆan bˆa´t di.nh: 1) d

Z

f(x)dx

=f(x)dx. 2)

Z

f(x)dx

0

=f(x). 3)

Z

df(x) =

Z

f0(x)dx=f(x) +C.

I

Z

0.dx=C. II

Z

1dx=x+C.

III

Z

xαdx = x α+1

α+ +C, α6=−1 IV

Z

dx

x = ln|x|+C, x6= V

Z

axdx= a x

lna +C (0 < a6= 1);

Z

exdx=ex+C. VI

Z

sinxdx=−cosx+C. VII

Z

cosxdx = sinx+C VIII

Z

dx

cos2x = tgx+C,x6=

π

2 +nπ,n ∈Z IX

Z dx

sin2x =−cotgx+C,x6=nπ, n∈Z X

Z

dx

√

1−x2 =

  

arc sinx+C,

−arc cosx+C

−1< x <1 XI

Z

dx +x2 =

  

arctgx+C,

−arccotgx+C. XII

Z

dx

√

x2±1 = ln|x+

√

x2±1|+C

(trong tru.`o.ng ho p dˆa´u tr`u th`ıx < −1 ho˘a.cx >1) XIII

Z

dx 1−x2 =

1 2ln

+1−xx

1)

Z

kf(x)dx=k

Z

f(x)dx, k6= 2)

Z

[f(x)±g(x)]dx=

Z

f(x)dx±

Z

g(x)dx.

3) Nˆe´u

Z

f(x)dx = F(x) +C v`a u = ϕ(x) kha’ vi liˆen tu.c th`ı

Z

f(u)du =F(u) +C.

C ´AC V´I DU.

V´ı du 1. Ch´u.ng minh r˘a`ng h`am y = signx c´o nguyˆen h`am trˆen khoa’ng bˆa´t k`y khˆong ch´u.a diˆe’mx= v`a khˆong c´o nguyˆen h`am trˆen mo.i khoa’ng ch´u.a diˆe’mx =

Gia’i. 1) Trˆen khoa’ng bˆa´t k`y khˆong ch´u.a diˆe’mx= h`amy= signx l`a h˘a`ng sˆo´ Ch˘a’ng ha.n v´o.i mo.i khoa’ng (a, b), 0< a < bta c´o signx = v`a d´o mo.i nguyˆen h`am cu’a n´o trˆen (a, b) c´o da.ng

F(x) =x+C, C ∈R

2) Ta x´et khoa’ng (a, b) m`a a < < b Trˆen khoa’ng (a,0) mo.i nguyˆen h`am cu’a signxc´o da.ngF(x) =−x+C1 c`on trˆen khoa’ng (0, b)

nguyˆen h`am c´o da.ng F(x) =x+C2 V´o.i mo.i c´ach cho.n h˘a`ng sˆo´C1

v`aC2 ta thu du.o c h`am [trˆen (a, b)] khˆong c´o da.o h`am ta.i diˆe’mx=

Nˆe´u ta cho.n C = C1 = C2 th`ı thu du.o c h`am liˆen tu.c y = |x| +C

nhu.ng khˆong kha’ vi ta.i diˆe’m x = T`u d´o, theo di.nh ngh˜ıa h`am signx khˆong c´o nguyˆen h`am trˆen (a, b),a <0< b. N

V´ı du 2. T`ım nguyˆen h`am cu’a h`am f(x) =e|x|trˆen to`

an tru.c sˆo´

Gia’i. V´o.i x > ta c´o e|x| = ex v`a d´o miˆ`ne x > mˆo.t c´ac nguyˆen h`am l`a ex Khi x < 0 ta c´o e|x| = e−x v`a vˆ

a.y miˆ`ne x < mˆo.t c´ac nguyˆen h`am l`a −e−x+C v´o.i h˘a`ng sˆo´C bˆa´t k`y

Theo di.nh ngh˜ıa, nguyˆen h`am cu’a h`am e|x| pha’i liˆ

pha’i tho’a m˜an diˆ`u kiˆe.ne lim x→0+0e

x

= lim x→0−0(

−e−x+C) t´u.c l`a =−1 +C ⇒C =

Nhu vˆa.y

F(x) =

        

ex nˆe´u x > 0, nˆe´u x= 0,

−e−x+ nˆe´u x <

l`a h`am liˆen tu.c trˆen to`an tru.c sˆo´ Ta ch´u.ng minh r˘a`ngF(x) l`a nguyˆen h`am cu’a h`am e|x| trˆen to`an tru.c sˆo´ Thˆa.t vˆa.y, v´o.i x > ta c´o F0(x) = ex = e|x|, v´o.i x <0 th`ıF0(x) = e−x = e|x| Ta c`on cˆ` n pha’ia ch´u.ng minh r˘a`ng F0(0) =e0 = Ta c´o

F+0(0) = lim

x→0+0

F(x)−F(0)

x = limx→0+0

ex−1 x = 1, F−0(0) = lim

x→0−0

F(x)−F(0)

x = limx→0−0

−e−x+ 2−1 x = Nhu vˆa.y F+0(0) =F

0

−(0) =F

(0) = = e|x| T`u d´o c´o thˆe’ viˆe´t:

Z

e|x|dx=F(x) +C =

  

ex+C, x <0

−e−x+ +C, x <0. N

V´ı du 3. T`ım nguyˆen h`am c´o dˆ` thi qua diˆe’m (o −2,2) dˆo´i v´o.i h`am f(x) =

x, x∈(−∞,0)

Gia’i. V`ı (ln|x|)0 =

x nˆen ln|x| l`a mˆo.t c´ac nguyˆen h`am cu’a h`am f(x) =

x Do vˆa.y, nguyˆen h`am cu’a f l`a h`am F(x) = ln|x|+C, C ∈ R H˘a`ng sˆo´ C du.o c x´ac di.nh t`u diˆe`u kiˆe.n F(−2) = 2, t´u.c l`a ln2 +C = 2⇒C = 2−ln2 Nhu vˆa.y

F(x) = ln|x|+ 2−ln2 = ln

V´ı du 4. T´ınh c´ac t´ıch phˆan sau dˆay: 1)

Z 2x+1−5x−1

10x dx, 2)

Z 2x+ 3

3x+ 2dx.

Gia’i. 1) Ta c´o I =

Z

2 x

10x − 5x 5·10x

dx=

Z h

21 x − 1 xi dx =

Z 1

5

x

dx−1

5

Z 1

2 x dx = 1 x ln1

5 − 1 x ln1 +C

=−

5xln5 +

5·2xln2 +C. 2)

I =

Z 2x+

2

3x+

dx=

3

h

x+2 +5 i

x+2

dx

= 3x+

5 9ln

x+2

3

+C.N

V´ı du 5. T´ınh c´ac t´ıch phˆan sau dˆay: 1)

Z

tg2xdx, 2)

Z

1 + cos2x

1 + cos 2xdx, 3)

Z √

1−sin 2xdx

Gia’i. 1)

Z

tg2xdx=

Z

sin2x cos2xdx=

Z

1−cos2x cos2x dx

=

Z

dx cos2x −

Z

2)

Z 1 + cos2x

1 + cos 2xdx=

Z 1 + cos2x

2 cos2x dx =

1

Z dx

cos2x+

Z

dx

=

2(tgx+x) +C. 3)

Z √

1−sin 2xdx =Z psin2x−2 sinxcosx+ cos2xdx

=Z p(sinx−cosx)2dx=

Z

|sinx−cosx|dx = (sinx+ cosx)sign(cosx−sinx) +C. N

B `AI T ˆA P

B˘a`ng c´ac ph´ep biˆe´n dˆo’i dˆo` ng nhˆa´t, h˜ay du.a c´ac t´ıch phˆan d˜a cho vˆ` t´ıch phˆan ba’ng v`a t´ınh c´ac t´ıch phˆan d´oe

1.

Z dx

x4−1 (DS

1 4ln

xx−+ 11−

2arctgx) 2.

Z 1 + 2x2

x2(1 +x2)dx. (DS arctgx−

1 x) 3.

Z √

x2+ +√1−x2

√

1−x4 dx. (DS arc sinx+ ln|x+

√

1 +x2|)

4.

Z √

x2+ 1−√1−x2

√

x4−1 dx (DS ln|x+

√

x2−1| −ln|x+√x2+ 1|)

5.

Z √

x4+x−4+ 2

x3 dx. (DS ln|x| −

1 4x4)

6.

Z

23x−1

ex−1 dx. (DS e2x

2 +e x+ 1)

1Dˆ

7.

Z

22x−1

√

2x dx. (DS ln2

h23x

2

3 + −x i ) 8. Z dx

x(2 + ln2x) (DS √ 2arctg lnx √ 2) 9.

Z √3

ln2x

x dx. (DS 5ln

5/3

x)

10.

Z

ex+e2x

1−ex dx. (DS −e x−

2ln|ex−1|) 11.

Z

exdx

1 +ex (DS ln(1 +e x))

12.

Z

sin2x

2dx. (DS 2x−

sinx ) 13.

Z

cotg2xdx. (DS −x−cotgx) 14.

Z √

1 + sin 2xdx,x∈

0,π

(DS −cosx+ sinx) 15.

Z

ecosxsinxdx. (DS −ecosx) 16.

Z

excosexdx. (DS sinex) 17.

Z

1

1 + cosxdx. (DS tg x 2) 18.

Z

dx

sinx+ cosx (DS √ 2ln tg x + π ) 19. Z

1 + cosx

(x+ sinx)3dx. (DS −

2

2(x+ sinx)2)

20.

Z

sin 2x

p

1−4 sin2x

dx. (DS −1

2

p

1−4 sin2x) 21.

Z

sinx

p

2−sin2x

dx. (DS −ln|cosx+

√

22.

Z sinxcosx p

3−sin4x

dx. (DS 2arc sin

sin2 x √ ) 23. Z arccotg3x

1 + 9x2 dx. (DS −

1 6arccotg 3x) 24. Z

x+√arctg2x

1 + 4x2 dx. (DS

1

8ln(1 + 4x

2) +

3arctg

3/22x)

25.

Z

arc sinx−arc cosx

√

1−x2 dx. (DS

1

2(arc sin

2

x+ arc cos2x))

26.

Z

x+ arc sin32x

√

1−4x2 dx. (DS −

1

√

1−4x2+1

8arc sin

4

2x)

27.

Z

x+ arc cos3/2x

√

1−x2 dx. (DS −

√

1−x2 −2

5arc cos

5/2

x)

28.

Z

x|x|dx. (DS |x|

3

3 ) 29.

Z

(2x−3)|x−2|dx. (DS F(x) =

     −2 3x

3+

2x

2 −6x+C, x < 2

2 3x

3−

2x

2+ 6x+C, x>2

)

30.

Z

f(x)dx, f(x) =

  

1−x2, |x|61, 1− |x|, |x|>1 (DS F(x) =

    

x− x

3

3 +C nˆe´u|x|61 x− x|x|

2 +

6signx+C nˆe´u|x|>1 )

1) H`am x=ϕ(t) x´ac di.nh v`a kha’ vi trˆen khoa’ngT v´o.i tˆa p ho..p gi´a tri l`a khoa’ng X.

2) H`am y=f(x)x´ac di.nh v`a c´o nguyˆen h`am F(x)trˆen khoa’ng X. Khi d´o h`am F(ϕ(t)) l`a nguyˆen h`am cu’a h`am f(ϕ(t))ϕ0(t) trˆen

khoa’ng T.

T`u di.nh l´y 10.1.1 suy r˘a`ng

Z

f(ϕ(t))ϕ0(t)dt=F(ϕ(t)) +C. (10.1) V`ı

F(ϕ(t)) +C = (F(x) +C)x=ϕ(t) =

Z

f(x)dxx=ϕ(t) cho nˆen d˘a’ng th´u.c (10.1) c´o thˆe’ viˆe´t du.´o.i da.ng

Z

f(x)dxx=ϕ(t)=

Z

f(ϕ(t))ϕ0(t)dt (10.2) D˘a’ng th´u.c (10.2) du.o c go.i l`a cˆong th´u.c dˆo’i biˆe´n t´ıch phˆan bˆa´t di.nh.

Nˆe´u h`am x = ϕ(t) c´o h`am ngu.o..c t = ϕ−1(x) th`ı t`u (10.2) thu

du.o c

Z

f(x)dx=

Z

f(ϕ(t))ϕ0(t)dtt=ϕ−1(x). (10.3)

Ta nˆeu mˆo.t v`ai v´ı du vˆe` ph´ep dˆo’i biˆe´n

i) Nˆe´u biˆe’u th´u.c du.´o.i dˆa´u t´ıch phˆan c´o ch´u.a c˘an

√

a2−x2,a >0

th`ı su.’ du.ng ph´ep dˆo’i biˆe´nx=asint, t∈−π

2, π

ii) Nˆe´u biˆe’u th´u.c du.´o.i dˆa´u t´ıch phˆan c´o ch´u.a c˘an

√

x2−a2,a >0

th`ı d`ung ph´ep dˆo’i biˆe´nx= a

cost, < t < π

2 ho˘a.cx=acht. iii) Nˆe´u h`am du.´o.i dˆa´u t´ıch phˆan ch´u.a c˘an th´u.c

√

a2+x2, a > 0

th`ı c´o thˆe’ d˘a.tx=atgt, t∈− π

2, π

ho˘a.cx =asht.

C ´AC V´I DU. V´ı du 1. T´ınh

Z

dx cosx

Gia’i. Ta c´o

Z

dx cosx =

Z

cosxdx

1−sin2x (d˘a.t t = sinx, dt= cosxdx) =

Z

dt 1−t2 =

1 2ln

+1−tt

+C = ln

tg x + π

+C. N

V´ı du 2. T´ınh I =

Z

x3dx

x8−2

Gia’i. ta c´o

I =

Z

4d(x

4

) x8−2 =

Z √ d x4 √

−2h1−x

4

√

2

2i

D˘a.t t= x

4

√

2 ta thu du.o c I =−

√ ln √

2 +x4

√

2−x4

+C. N

V´ı du 3. T´ınh I =

Z

x2dx

p

(x2+a2)3 ·

Gia’i. D˘a.t x(t) =atgt⇒dx= adt

cos2t Do d´o

I =

Z

a3tg2t·cos3tdt a3cos2t =

Z

sin2t costdt =

Z

dt cost −

Z

costdt = ln

tgt

2 + π

−sint+C.

V`ıt= arctgx a nˆen I = ln

tg 1 2arctg x a + π −sin arctgx a +C =−√ x

x2+a2 + ln|x+

√

Thˆa.t vˆa.y, v`ı sinα = cosα·tgα nˆen dˆ˜ d`ang thˆa´y r˘a`nge sin arctgx a

= √ x

x2+a2 ·

Tiˆe´p theo ta c´o sin1

2arctg x a + π

cos1 2arctg x a + π =

1−cosarctgx a + π sin arctgx a + π =

1 + sinarctgx a −cos arctgx a

= x+

√

a2+x2

a

v`a t`u d´o suy diˆ`u pha’i ch´e u.ng minh N

V´ı du 4. T´ınh I =

Z √

a2+x2dx.

Gia’i. D˘a.t x=asht Khi d´o I =

Z q

a2(1 + sh2

t)achtdt=a2

Z

ch2tdt =a2

Z

ch2t+ dt=

a2

2

1

2sh2t+t

+C = a

2

2(sht·cht+t) +C. V`ı cht = p1 + sh2t =

r

1 +x

2

a2 e

t

= sht+ cht = x+

√

a2+x2

a nˆen t= ln

x+

√

a2+x2

a

v`a d´o

Z √

a2+x2dx= x

2

√

a2+x2+a

2 ln|x+

√

a2+x2|+C. N

V´ı du 5. T´ınh 1) I1 =

Z

x2+

√

x6−7x4+x2dx, 2) I2 =

Z

3x+

√

Gia’i. 1) Ta c´o

I1 =

Z +

x2

r

x2 −7 +

x2

dx=

Z d

x−

x

r

x−

x −5 = Z dt √

t2−5

= ln|t+

√

t2−5|+C = ln

x−

x +

r

x2−7 +

x2

+C.

2) Ta viˆe´t biˆe’u th´u.c du.´o.i dˆa´u t´ıch phˆan du.´o.i da.ng f(x) =−3

2·

−2x+

√

−x2+ 6x−8 + 13·

1

√

−x2+ 6x−8

v`a thu du.o..c I2 =

Z

f(x)dx =−3

2

Z

(−x2+ 6x−8)−12d(−x2+ 6x−8) + 13

Z d(x−3)

p

1−(x−3)2

=−3

√

−x2+ 6x−8 + 13 arc sin(x−3) +C. N

V´ı du 6. T´ınh 1)

Z dx

sinx, 2) I2 =

Z sinxcos3x

1 + cos2xdx.

Gia’i

1) C´ach I Ta c´o

Z

dx sinx =

Z

sinx sin2xdx=

Z

d(cosx) cos2x−1 =

1 2ln

1−cosx + cosx +C.

C´ach II

Z

dx sinx =

Z d x sinx 2cos x = Z d x tgx ·cos

2) Ta c´o

I2 =

Z

sinxcosx[(cos2x+ 1)−1]

1 + cos2x dx.

Ta d˘a.t t= + cos2x T`u d´o dt=−2 cosxsinxdx Do d´o

I2 =−

1

Z

t−1

t dt =− t

2 + ln|t|+C, d´o t= + cos2x. N

V´ı du 7. T´ınh 1) I1 =

Z

exdx

√

e2x+ 5 , 2) I2 =

Z

ex+ 1 ex−1dx.

Gia’i

1) D˘a.tex =t Ta c´oexdx =dt v`a I1 =

Z

dt

√

t2+ 5 = ln|t+

√

t2 + 5|+C = ln|ex +

√

e2x+ 5|+C. 2) Tu.o.ng tu , d˘a.t ex =t, exdx=dt,dx = dt

t v`a thu du.o c I2 =

Z t+ 1

t−1 dt

t =

Z 2dt

t−1−

Z dt

t = 2ln|t−1| −ln|t|+C = 2ln|ex−1| −lnex+c

= ln(ex−1)2 −x+C. N

B `AI T ˆA P

T´ınh c´ac t´ıch phˆan: 1.

Z

e2x

4

√

ex+ 1dx. (DS 21(3e

x− 4)p4

(ex+ 1)3)

2.

Z

dx

√

ex+ 1 (DS ln

√

1 +ex−1

√

1 +ex+ 1

)

3.

Z

e2x

ex−1dx. (DS e x

+ ln|ex−1|) 4.

Z √

1 + lnx

x dx. (DS

p

(1 + lnx)3)

5.

Z √

1 + lnx xlnx dx. (DS

√

1 + lnx−ln|lnx|+ 2ln| √

1 + lnx−1|) 6.

Z

dx

ex/2+ex (DS −x−2e

−x2 + 2ln(1 +ex2)) 7.

Z

arctg√x

√

x

dx

1 +x (DS (arctg

√

x)2) 8.

Z √

e3x+e2xdx. (DS. 3(e

x+ 1)3/2)

9.

Z

e2x2+2x−1(2x+ 1)dx (DS 2e

2x2+2x−1

) 10.

Z

dx

√

ex−1 (DS 2arctg

√

ex−1) 11.

Z

e2xdx

√

e4x+ 1 (DS 2ln(e

2x+√e4x+ 1)) 12.

Z 2xdx

√

1−4x (DS

arc sin 2x ln2 ) 13.

Z

dx

1 +√x+ (DS 2[

√

x+ 1−ln(1 +√x+ 1)])

Chı’ dˆa˜n. D˘a.t x+ =t2 14.

Z

x+

x√x−2dx. (DS

√

x−2 +

√

2arctg

r

x−2 ) 15.

Z

dx

√

ax+b+m (DS a

√

ax+b−mln| √

16.

Z

dx

3

√

x(√3x−1) (DS 3

√

x+ 3ln|√3x−1|)

17.

Z

dx

(1−x2)3/2 (DS tg(arc sinx))

Chı’ dˆa˜n. D˘a.t x= sint, t∈− π

2, π ) 18. Z dx

(x2+a2)3/2 (DS

1 a2 sin

arctgx a

)

Chı’ dˆa˜n. D˘a.t x=atgt, t∈− π

2, π 19. Z dx

(x2−1)3/2 (DS.−

1

cost, t= arc sin x)

Chı’ dˆa˜n. D˘a.t x= sint,−

π

2 < t <0, 0< t < π 20.

Z √

a2−x2dx. (DS. a

2arc sin x a +

x√a2−x2

2 )

Chı’ dˆa˜n. D˘a.t x=asint. 21.

Z √

a2+x2dx. (DS. x

2

√

a2+x2+a

2ln|x+

√

a2+x2|)

Chı’ dˆa˜n. D˘a.t x=asht. 22.

Z

x2

√

a2+x2dx. (DS

1

x

√

a2+x2−a2

ln(x+

√

a2+x2))

23.

Z

dx

x2√x2+a2 (DS −

√

x2+a2

a2x )

Chı’ dˆa˜n. D˘a.t x=

t ho˘a.c x=atgt, ho˘a.c x=asht. 24.

Z x2dx

√

a2−x2 (DS

a2

2arc sin x a −

x a

√

a2−x2)

Chı’ dˆa˜n. D˘a.t x=asint. 25.

Z

dx x

√

x2−a2 (DS −

1 aarc sin

Chı’ dˆa˜n. D˘a.t x=

t, ho˘a.c x= a

cost ho˘a.c x=acht. 26.

Z √

1−x2

x2 dx. (DS −

√

1−x2

x −arc sinx) 27.

Z

dx

p

(a2+x2)3 (DS

x a2√x2+a2)

28.

Z dx

x2√x2−9 (DS

√

x2−9

9x ) 29.

Z

dx

p

(x2−a2)3 (DS −

x a2√x2−a2)

30.

Z

x2

√

a2 −x2dx.

(DS − x

4(a

2−

x2)3/2+a

2

8x

√

x2−a2+a

8arc sin x a)

31.

Z r

a+x

a−xdx. (DS −

√

a2−x2+ arc sinx

a)

Chı’ dˆa˜n. D˘a.t x=acos 2t 32.

Z r

x−a x+adx. (DS

√

x2−a2−2aln(√x−a+√x+a) nˆe´u x > a,

− √

x2−a2+ 2aln(√−x+a+√−x−a) nˆe´u x <−a)

Chı’ dˆa˜n. D˘a.t x= a cos 2t 33.

Z r

x−1 x+

dx

x2 (DS arc cos

1 x−

√

x2−1

x )

Chı’ dˆa˜n. D˘a.t x= t 34.

Z

dx

√

x−x2 (DS 2arc sin

√

Chı’ dˆa˜n. D˘a.t x= sin2t. 35.

Z √

x2+ 1

x dx. (DS

√

x2+ 1−ln

+

√

x2+ 1

x

) 36.

Z

x3dx

√

2−x2 (DS −

x2

3

√

2−x2−

3

√

2−x2)

37.

Z p

(9−x2)2

x6 dx. (DS −

p

(9−x2)5

45x5 )

38.

Z

x2dx

√

x2−a2 (DS

x

√

x2−a2+a

2ln|x+

√

x2−a2|)

39.

Z

(x+ 1)dx

x(1 +xex) (DS ln

xex

1 +xex

)

Chı’ dˆa˜n. Nhˆan tu.’ sˆo´ v`a mˆ˜u sˆo´ v´o.ia ex rˆ` i d˘a.to xex=t. 40.

Z

dx

(x2+a2)2 (DS

1 2a3

h

arctgx a +

ax x2+a2

i

)

Chı’ dˆa˜n. D˘a.t x=atgt.

10.1.3 Phu.o.ng ph´ap t´ıch phˆan t`u.ng phˆ` na Phu.o.ng ph´ap t´ıch phˆan t`u.ng phˆ` n du a trˆen di.nh l´y sau dˆay.a

D- i.nh l´y. Gia’ su.’ trˆen khoa’ngD c´ac h`am u(x)v`a v(x)kha’ vi v`a h`am

v(x)u0(x) c´o nguyˆen h`am Khi d´o h`am u(x)v0(x) c´o nguyˆen h`am trˆen D v`a

Z

u(x)v0(x)dx=u(x)v(x)−

Z

v(x)u0(x)dx (10.4) Cˆong th´u.c (10.4) du.o c go.i l`a cˆong th´u.c t´ınh t´ıch phˆan t`u.ng phˆa` n V`ıu0(x)dx=du v`a v0(x)dx=dv nˆen (10.4) c´o thˆe’ viˆe´t du.´

o.i da.ng

Z

udv=uv−

Z

vdu. (10.4*)

Nh´om Igˆ` m nh˜o u.ng t´ıch phˆan m`a h`am du.´o.i dˆa´u t´ıch phˆan c´o ch´u.a th`u.a sˆo´ l`a mˆo.t c´ac h`am sau dˆay: lnx, arc sinx, arc cosx, arctgx, (arctgx)2, (arc cosx)2, lnϕ(x), arc sinϕ(x),

Dˆe’ t´ınh c´ac t´ıch phˆan n`ay ta ´ap du.ng cˆong th´u.c (10.4*) b˘a`ng c´ach d˘a.tu(x) b˘a`ng mˆo.t c´ac h`am d˜a chı’ c`ondv l`a phˆ` n c`on la.i cu’aa biˆe’u th´u.c du.´o.i dˆa´u t´ıch phˆan

Nh´om II gˆ` m nh˜o u.ng t´ıch phˆan m`a biˆe’u th´u.c du.´o.i dˆa´u t´ıch phˆan c´o da.ng P(x)eax, P(x) cosbx,P(x) sinbxtrong d´o P(x) l`a da th´u.c, a, bl`a h˘a`ng sˆo´

Dˆe’ t´ınh c´ac t´ıch phˆan n`ay ta ´ap du.ng (10.4*) b˘a`ng c´ach d˘a.tu(x) = P(x), dv l`a phˆ` n c`on la.i cu’a biˆe’u th´u.c du.´o.i dˆa´u t´ıch phˆan Sau mˆo˜ia lˆ` n t´ıch phˆan t`a u.ng phˆ` n bˆa.c cu’a da th´u.c s˜e gia’m mˆo.t do.n vi a

Nh´om III gˆ` m nh˜o u.ng t´ıch phˆan m`a h`am du.´o.i dˆa´u t´ıch phˆan c´o da.ng: eaxsinbx, eaxcosbx, sin(lnx), cos(lnx), Sau hai lˆ` n t´ıch phˆana t`u.ng phˆ` n ta la.i thu du.o c t´ıch phˆan ban dˆaa ` u v´o.i hˆe sˆo´ n`ao d´o D´o l`a phu.o.ng tr`ınh tuyˆe´n t´ınh v´o.i ˆa’n l`a t´ıch phˆan cˆ` n t´ınh.a

Du.o.ng nhiˆen l`a ba nh´om v`u.a nˆeu khˆong v´et hˆe´t mo.i t´ıch phˆan t´ınh du.o c b˘a`ng t´ıch phˆan t`u.ng phˆa` n (xem v´ı du 6)

Nhˆa n x´et. Nh`o c´ac phu.o.ng ph´ap dˆo’i biˆe´n v`a t´ıch phˆan t`u.ng phˆ` na ta ch´u.ng minh du.o c c´ac cˆong th´u.c thu.`o.ng hay su.’ du.ng sau dˆay:

1)

Z

dx x2+a2 =

1 aarctg

x

a +C, a6= 2)

Z

dx a2−x2 =

1 2aln

a+x

a−x

+C, a6= 3)

Z

dx

√

a2−x2 = arc sin

x

a +C, a 6= 4)

Z

dx

√

x2±a2 = ln|x+

√

C ´AC V´I DU. V´ı du 1. T´ınh t´ıch phˆan I =

Z √

xarctg√xdx.

Gia’i. T´ıch phˆan d˜a cho thuˆo.c nh´om I Ta d˘a.t u(x) = arctg√x,

dv =√xdx. Khi d´o du=

1 +x · dx 2√x, v=

2 3x

3

2 Do d´o

I = 3x

3 2arctg

√

x−

3

Z x

1 +xdx =

3x

3 2arctg

√

x−

3

Z h

1−

1 +x

i

dx =

3x

3 arctg

√

x−

3(x−ln|1 +x|) +C. N V´ı du 2. T´ınh I =

Z

arc cos2xdx.

Gia’i. Gia’ su.’ u= arc cos2x, dv=dx Khi d´o

du=−2arc cos√ x

1−x2 dx, v=x.

Theo (10.4*) ta c´o

I =xarc cos2x+

Z xarc cosx

√

1−x2 dx.

Dˆe’ t´ınh t´ıch phˆan o.’ vˆe´ pha’i d˘a’ng th´u.c thu du.o..c ta d˘a.t u = arc cosx,dv= √xdx

1−x2 Khi d´o

du=−√ dx

1−x2 , v=−

Z

d(

√

1−x2) =−

√

1−x2+C

v`a ta chı’ cˆ` n lˆa´ya v =− √

1−x2:

Z

xarc cosx

√

21−x2dx=−

√

1−x2arc cosx−

Z

dx =−

√

Cuˆo´i c`ung ta thu du.o..c I =xarc cos2x−2

√

1−x2arc cosx−2x+C. N

V´ı du 3. T´ınh I =

Z

x2sin 3xdx

Gia’i. T´ıch phˆan d˜a cho thuˆo.c nh´om II Ta d˘a.t u(x) = x2,

dv = sin 3xdx Khi d´o du= 2xdx, v=−1

3cos 3x v`a I =−1

3x

2

cos 3x+2

Z

xcos 3xdx =−1

3x

2

cos 3x+ 3I1.

Ta cˆ` n t´ınha I1 D˘a.t u = x, dv = cos 3xdx Khi d´o du = 1dx,

v=

3sin 3x T`u d´o I =−1

3x

2

cos 3x+

h1

3xsin 3x−

Z

sin 3xdx

i

=−1

3x

2

cos 3x+

9xsin 3x+

27cos 3x+C. N

Nhˆa n x´et. Nˆe´u d˘a.t u = sin 3x, dv = x2dx th`ı lˆ` n t´ıch phˆan t`a u.ng phˆ` n th´a u nhˆa´t khˆong du.a dˆe´n t´ıch phˆan do.n gia’n ho.n

V´ı du 4. T´ınh I =

Z

eaxcosbx; a, b6=

Gia’i. Dˆay l`a t´ıch phˆan thuˆo.c nh´om III Ta d˘a.t u = eax, dv = cosbxdx Khi d´o du=aeaxdx,v =

bsinbx v`a I =

be ax

sinbx− a

b

Z

eaxsinbxdx= be

ax

sinbx− a

bI1.

Dˆe’ t´ınh I1 ta d˘a.t u = eax, dv = sinbxdx Khi d´o du = aeaxdx,

v=−1

bcosbxv`a I1 =−

1 be

ax

cosbx+a b

Z

Thˆe´I1 v`ao biˆe’u th´u.c dˆo´i v´o.i I ta thu du.o c

Z

eaxcosbxdx= be

ax

sinbx+ a

b2 cosbx−

a2

b2

Z

eaxcosbxdx. Nhu vˆa.y sau hai lˆa` n t´ıch phˆan t`u.ng phˆ` n ta thu du.o c phu.o.nga tr`ınh tuyˆe´n t´ınh v´o.i ˆa’n l`aI Gia’i phu.o.ng tr`ınh thu du.o c ta c´o

Z

eaxcosbxdx=eaxacosbx+bsinbx

a2+b2 +C. N

V´ı du 5. T´ınh I =R sin(ln x)dx

Gia’i. D˘a.t u = sin(lnx), dv = dx Khi d´o du =

xcos(lnx)dx, v=x Ta thu du.o c

I =xsin(lnx)−

Z

cos(lnx)dx=xsin(lnx)−I1.

Dˆe’ t´ınh I1 ta la.i d˘a.t u = cos(lnx), dv = dx. Khi d´o du =

−1

xsin(lnx)dx,v =x v`a

I1 =xcos(lnx) +

Z

sin(lnx)dx

Thay I1 v`ao biˆe’u th´u.c dˆo´i v´o.i I thu du.o c phu.o.ng tr`ınh

I =x(sin lnx−cos lnx)−I v`a t`u d´o

I = x

2(sin lnx−cos lnx) +C. N

V´ı du 6. T´ınh 1) I =

Z

xdx

sin2x; 2) In =

Z

dx

(x2+a2)n, n ∈N

Gia’i. 1) R˜o r`ang t´ıch phˆan n`ay khˆong thuˆo.c bˆa´t c´u nh´om n`ao ba nh´om d˜a nˆeu Thˆe´ nhu.ng b˘a`ng c´ach d˘a.tu =x,dv = dx

sin2x v`a ´ap du.ng cˆong th´u.c t´ıch phˆan t`u.ng phˆa` n ta c´o

I =−xcotgx+

Z

cotgxdx =−xcotgx+

Z

cosx sinxdx =−xcotgx+

Z

d(sinx)

sinx =−xcotgx+ ln|sinx|+C. 2) T´ıch phˆan In du.o..c biˆe’u diˆe˜n du.´o.i da.ng

In= a2

Z

x2+a2−x2

(x2+a2)n dx= a2

h Z dx

(x2+a2)n−1 −

Z

x2dx

(x2 +a2)n

i

=

a2In−1−

1 2a2

Z

x 2xdx (x2+a2)n·

Ta t´ınh t´ıch phˆan o.’ vˆe´ pha’i b˘a`ng phu.o.ng ph´ap t´ıch phˆan t`u.ng phˆ` n D˘a.ta u = x, dv = 2xdx

(x2+a2)n =

d(x2+a2)

(x2+a2)n Khi d´o du = dx, v =−

(n−1)(x2+a2)n−1 v`a

1 2a2

Z

x 2xdx (x2+a2)n =

−x

2a2(n−1)(x2+a2)n−1 +

1

2a2(n−1)In−1

T`u d´o suy r˘a`ng In=

a2In−1+

x

2a2(n−1)(x2 +a2)n−1 −

1

2a2(n−1)In−1

hay l`a

In= x

2a2(n−1)(x2+a2)n−1 +

2n−3

Ta nhˆa.n x´et r˘a`ng t´ıch phˆanInkhˆong thuˆo.c bˆa´t c´u nh´om n`ao ba nh´om d˜a chı’

Khi n = ta c´o I1 =

Z

dx x2+a2 =

1 aarctg

x a +C. ´

Ap du.ng cˆong th´u.c truy hˆo` i (*) ta c´o thˆe’ t´ınhI2 qua I1 rˆ` io I3 qua

I2, N

V´ı du 7. T´ınh I =

Z

xeaxcosbxdx.

Gia’i. D˘a.t u=x, dv=eaxcosbxdx Khi d´o du=dx, v=eaxacosbx+bsinbx

a2+b2

(xem v´ı du 4) Nhu vˆa.y I =xeaxacosbx+bsinbx

a2+b2 −

1 a2+b2

Z

eax(acosbx+bsinbx)dx =xeaxacosbx+bsinbx

a2+b2 −

a a2+b2

Z

eaxcosbxdx

− b

a2+b2

Z

eaxsinbxdx.

T´ıch phˆan th´u nhˆa´t o.’ vˆe´ pha’i du.o c t´ınh v´ı du 4, t´ıch phˆan th´u hai du.o c t´ınh tu.o.ng tu v`a b˘a`ng

Z

eaxsinbxdx=eaxasinbx−bcosbx a2+b2 ·

Thay c´ac kˆe´t qua’ thu du.o..c v`ao biˆe’u th´u.c dˆo´i v´o.i I ta c´o I = e

ax

a2+b2

h

x− a

a2+b2

(acosbx+bsinbx)

− b

a2+b2(asinbx−bcosbx)

i

+C N

1.

Z

x2xdx. (DS

x(xln 2−1) ln22 ) 2.

Z

x2e−xdx. (DS −x2e−x−2xe−x−2e−x) 3.

Z

x3e−x2dx. (DS −1

2(x

2 + 1)e−x2

) 4.

Z

(x3+x)e5xdx. (DS 5e

5x x3−

5x

2

+31 25x−

31 125 ) 5. Z

arc sinxdx. (DS xarc sinx+√1−x2)

6.

Z

xarc sinxdx. (DS 4(2x

2−

1)arc sinx+ 4x

√

1−x2)

7.

Z

x2arc sin 2xdx (DS x

3

3arc sin 2x+

2x2+ 1

36

√

1−4x2)

8.

Z

arctgxdx (DS xarctgx−

2ln(1 +x

2))

9.

Z

arctg√xdx. (DS (1 +x)arctg√x−√x) 10.

Z

x3arctgxdx (DS x

4−

1

4 arctgx− x3 12 + x 4) 11. Z

(arctgx)2xdx (DS. x

2+ 1

2 (arctgx)

2−xarctgx+

2ln(1 +x

2))

12.

Z

(arc sinx)2dx. (DS x(arc sinx)2+ 2arc sinx

√

1−x2−2x)

13.

Z

arc sinx

√

x+ 1dx. (DS

√

x+ 1arc sinx+ 4√1−x) 14.

Z

arc sinx

x2 dx. (DS −

arc sinx x −ln

+

√

1−x2

x ) 15. Z xarctgx √

1 +x2dx. (DS

√

16.

Z

arc sin√x

√

1−x dx. (DS 2(

√

x−√1−xarc sin√x)) 17.

Z

lnxdx. (DS x(lnx−1)) 18.

Z √

xln2xdx. (DS 3x

3/2ln2

x−

3lnx+

) 19.

Z

ln(x+

√

16 +x2)dsx (DS. xln(x+√16 +x2)−√16 +x2)

20.

Z

xln(x+

√

1 +x2)

√

1 +x2 dx. (DS

√

1 +x2ln(x+√1 +x2)−x)

21.

Z

sinxln(tgx)dx (DS lntgx

−cosxln(tgx)) 22.

Z

x2ln(1 +x)dx. (DS (x

3+ 1) ln(x+ 1)

3 −

x3

9 + x2

6 − x 3) 23.

Z

x2sin 2xdx (DS 1−2x

2

4 cos 2x+ x

2sin 2x) 24.

Z

x3cos(2x2)dx (DS 8(2x

2

sin 2x2+ cos 2x2)) 25.

Z

exsinxdx. (DS e

x(sinx−cosx)

2 )

26.

Z

3xcosxdx. (DS sinx+ (ln 3) cosx + ln23

x)

27.

Z

e3x(sin 2x−cos 2x)dx (DS e

3x

13(sin 2x−5 cos 2x)) 28.

Z

xe2xsin 5xdx

(DS e

2x 29

h

2x+ 21 29

sin 5x+−5x+20 29

cos 5xi)

29.

Z

x2exsinxdx. (DS

30.

Z

x2excosxdx. (DS (x−1)

2

sinx+ (x2−1) cosx

2 e

x ) 31.

Z

x2sin(lnx)dx. (DS [3 sinx(lnx)−cos(lnx)]x

3

10 )

32. T`ım cˆong th´u.c truy hˆ` i dˆo´i v´o.i mˆo˜i t´ıch phˆano In du.o c cho du.´o.i dˆay:

1) In =

Z

xneaxdx, a6= (DS In= ax

n

eax− n

aIn−1) 2) In =

Z

lnnxdx. (DS In=xlnnx−nIn−1)

3) In=

Z

xαlnnxdx,α 6=−1 (DS In= x α+1

lnnx α+ −

n

α+ 1In−1) 4)In =

Z

xndx

√

x2+a,n >2 (DS.In=

xn−1√x2+a

n −

n−1 n aIn−2) 5)In=

Z

sinnxdx,n >2 (DS.In=−cosxsin

n−1

x

n +

n−1 n In−2) 6) In =

Z

cosnxdx, n >2 (DS In= sinxcos n−1x

n +

n−1 n In−2) 7) In=

Z

dx

cosnx, n >2 (DS.In =

sinx

(n−1) cosn−1x+

n−2 n−1In−2) 10.2 C´ac l´o.p h`am kha’ t´ıch l´o.p c´ac

h`am so cˆa´p

10.2.1 T´ıch phˆan c´ac h`am h˜u.u ty’ 1) Phu.o.ng ph´ap hˆe sˆo´ bˆa´t di.nh H`am da.ng

trong d´oPm(x) l`a da th´u.c bˆa.cm,Qn(x) l`a da th´u.c bˆa.cn du.o..c go.i l`a h`am h˜u.u ty’ (hay phˆan th´u.c h˜u.u ty’) Nˆe´u m > n th`ıPm(x)/Qn(x) du.o c go.i l`a phˆan th´u.c h˜u.u ty’ khˆong thu c su ; nˆe´u m < n th`ı Pm(x)/Qn(x) du.o c go.i l`a phˆan th´u.c h˜u.u ty’ thu c su

Nˆe´u R(x) l`a phˆan th´u.c h˜u.u ty’ khˆong thu c su th`ı nh`o ph´ep chia da th´u.c ta c´o thˆe’ t´ach phˆ` n nguyˆena W(x) l`a da th´u.c cho

R(x) = Pm(x)

Qn(x) =W(x) + Pk(x)

Qn(x) (10.5) d´o k < nv`aW(x) l`a da th´u.c bˆa.cm−n.

T`u (10.5) suy r˘a`ng viˆe.c t´ınh t´ıch phˆan phˆan th´u.c h˜u.u ty’ khˆong thu c su du.o c quy vˆe` t´ınh t´ıch phˆan phˆan th´u.c h˜u.u ty’ thu c su v`a t´ıch phˆan mˆo.t da th´u.c.

D- i.nh l´y 10.2.1. Gia’ su.’ Pm(x)/Qn(x) l`a phˆan th´u.c h˜u.u ty’ thu c su v`a

Q(x) = (x−a)α· · ·(x−b)β(x2+px+q)γ· · ·(x2+rx+s)δ

trong d´o a, , b l`a c´ac nghiˆe.m thu c, x2 +px+q, , x2+rx+s l`a nh˜u.ng tam th´u.c bˆa c hai khˆong c´o nghiˆe.m thu..c Khi d´o

P(x) Q(x) =

Aα

(x−a)α +· · ·+ A1

x−a +· · ·+ Bβ (x−b)β +

Bβ−1

(x−b)β−1 +· · ·+

+ B1 x−b+

Mγx+Nγ

(x2 +px+q)γ +· · ·+

M1x+N1

x2+px+q +· · ·+

+ Kδx+Lδ

(x2+rx+s)δ +· · ·+

K1x+L1

x2+rx+s, (10.6)

trong d´o Ai, Bi, Mi, Ni, Ki v`a Li l`a c´ac sˆo´ thu..c.

C´ac phˆan th´u.c o.’ vˆe´ pha’i cu’a (10.6) du.o c go.i l`a c´ac phˆan th´u.c do.n gia’n hay c´ac phˆan th´u.c co ba’n v`a d˘a’ng th´u.c (10.6) du.o c go.i l`a khai triˆe’n phˆan th´u.c h˜u.u ty’ thu c su P(x)/Q(x) th`anh tˆo’ng c´ac phˆan th´u.c co ba’n v´o.i hˆe sˆo´ thu..c

Phu.o.ng ph´ap I. Quy dˆ` ng mˆa˜u sˆo´ d˘a’ng th´o u.c (10.6) v`a sau d´o cˆan b˘a`ng c´ac hˆe sˆo´ cu’a l˜uy th`u.a c`ung bˆa.c cu’a biˆe´nxv`a di dˆe´n hˆe phu.o.ng tr`ınh dˆe’ x´ac di.nh Ai, , Li (phu.o.ng ph´ap hˆe sˆo´ bˆa´t di.nh).

Phu.o.ng ph´ap II. C´ac hˆe sˆo´Ai, , Li c˜ung c´o thˆe’ x´ac di.nh b˘a`ng c´ach thayxtrong (10.6) (ho˘a.c d˘a’ng th´u.c tu.o.ng du.o.ng v´o.i (10.6)) bo.’i c´ac sˆo´ du.o..c cho.n mˆo.t c´ach th´ıch ho p

T`u (10.6) ta c´o

D- i.nh l´y 10.2.2. T´ıch phˆan bˆa´t di.nh cu’a mo.i h`am h˜u.u ty’ dˆe`u biˆe’u diˆ˜n du.o c qua c´ac h`am so cˆa´p m`a cu thˆe’ l`a qua c´ac h`am h˜u.u ty’, h`ame lˆogarit v`a h`am arctang.

C ´AC V´I DU. V´ı du 1. T´ınh I =

Z

xdx (x−1)(x+ 1)2

Gia’i. Ta c´o x

(x−1)(x+ 1)2 =

A x−1 +

B1

x+ + B2

(x+ 1)2

T`u d´o suy r˘a`ng

x=A(x+ 1)2+B1(x−1)(x+ 1) +B2(x−1) (10.7)

Ta x´ac di.nh c´ac hˆe sˆo´A,B1,B2 b˘a`ng c´ac phu.o.ng ph´ap sau dˆay

Phu.o.ng ph´ap I. Viˆe´t d˘a’ng th´u.c (10.7) du.´o.i da.ng x≡(A+B1)x2+ (2A+B2)x+ (A−B1−B2)

Cˆan b˘a`ng c´ac hˆe sˆo´ cu’a l˜uy th`u.a c`ung bˆa.c cu’a x ta thu du.o..c

    

A+B1=

2A+B2 =

A−B1−B2 =

T`u d´o A=

4, B1 =−

Phu.o.ng ph´ap II. Thayx = v`ao (10.7) ta c´o =A·4⇒A= Tiˆe´p theo, thay x = −1 v`ao (10.7) ta thu du.o c: −1 = −B2 ·2 hay

l`a B2 =

1

2 Dˆe’ t`ım B1 ta thˆe´ gi´a tri x = v`ao (10.7) v`a thu du.o c =A−B1−B2 hay l`aB1 =A−B2 =−

1 Do d´o

I =

Z

dx x−1−

1

Z

dx x+ +

1

Z

dx (x+ 1)2

=−

2(x+ 1) + 4ln

xx−+ 11+C. N

V´ı du 2. T´ınh I =

Z

3x+ x(1 +x2)2dx.

Gia’i. Khai triˆe’n h`am du.´o.i dˆa´u t´ıch phˆan th`anh tˆo’ng c´ac phˆan th´u.c co ba’n

3x+ x(1 +x2)2 =

A x +

Bx+C +x2 +

Dx+F (1 +x2)2

T`u d´o

3x+ 1≡(A+B)x4+Cx3+ (2A+B+D)x2+ (C+F)x+A. Cˆan b˘a`ng c´ac hˆe sˆo´ cu’a c´ac l˜uy th`u.a c`ung bˆa.c cu’a x ta thu du.o c

              

A+B = C =

2A+B+D = ⇒A = 1, B =−1, C = 0, D =−1, F = C +F =

A= T`u d´o suy r˘a`ng

I =

Z

dx x −

Z

xdx +x2 −

Z

xdx

(1 +x2)2 +

Z

dx (1 +x2)2

= ln|x| −1

2ln(1 +x

2

)−1

2(1 +x

2

)−2d(1 +x2) +

Z

dx (1 +x2)2

= ln|x| −1

2ln(1 +x

2

) +

Ta t´ınh I2 =

Z

dx

(1 +x2)2 b˘a`ng cˆong th´u.c truy hˆ` i thu du.o c trongo

10.1 Ta c´o I2 =

1 2·

x +x2 +

1 2I1 =

x 2(1 +x2)+

1

Z dx

1 +x2

= x

2(1 +x2)+

1

2arctgx+C. Cuˆo´i c`ung ta thu du.o c

I = ln|x| −

2ln(1 +x

2

) + 3x+ 2(1 +x2)+

3

2arctgx+C. N

B `AI T ˆA P T´ınh c´ac t´ıch phˆan (1-12)

1.

Z

xdx

(x+ 1)(x+ 2)(x−3) (DS

4ln|x+ 1| −

5ln|x+ 2|+

20|x−3|) 2.

Z 2x4 + 5x2−2

2x3 −x−1 dx.

DS x

2

2 + ln|x−1|+ ln(2x

2

+ 2x+ 1) + arctg(2x+ 1)) 3.

Z

2x3+x2+ 5x+ (x2+ 3)(x2−x+ 1)dx.

DS √1

3arctg x

√

3+ ln(x

2

−x+ 1) + √2

3arctg

2x−1

√

3 ) 4.

Z

x4+x2+ x(x−2)(x+ 2)dx.

(DS x

2

2 −

4ln|x|+ 21

8 ln|x−2|+ 21

5.

Z

dx

x(x−1)(x2−x+ 1)2

(DS ln

x−1

x − 10 √ 3arctg

2x−1

√

3

−

3

2x−1 x2−x+ 1)

6.

Z

x4 −x2+ 1

(x2−1)(x2+ 4)(x2−2)dx.

(DS −

10ln

xx−+ 11

+ 20arctg x 2+ √ 2ln x− √ x+ √ ) 7. Z

3x2+ 5x+ 12 (x2+ 3)(x2+ 1)dx.

(DS −

4ln(x

2

+ 3)− √ arctg x √ + 4ln(x

+ 1) +9

2arctgx) 8.

Z

(x4+ 1)dx x5+x4 −x3−x2

(DS ln|x|+ x+

1

2ln|x−1| −

2ln|x+ 1|+ x+ 1) 9.

Z

x3+x+ x4−1 dx.

(DS

4ln|x−1|+

4ln|x+ 1| −

2arctgx) 10.

Z

x4

1−x4dx.

(DS −x+ ln

xx+ 1−1

+1 2arctgx) 11. Z

3x+

(x2+ 2x+ 2)2dx.

(DS 2x−1

12.

Z

x4−2x2+ (x2−2x+ 2)2dx.

(DS x+ 3−x

x2−2x+ 2 + ln(x

−2x+ 2) + arctg(x−1)) 13.

Z

x2+ 2x+ 7

(x−2)(x2+ 1)3dx.

(DS 5ln|x

2

−2| −

10ln|x

2

+ 1|+ 1−x x2+ 1 −

11

5 arctgx) 14.

Z

x2

(x+ 2)2(x+ 1)dx.

(DS

x+ + ln|x+ 1|) 15.

Z x2+ 1

(x−1)3(x+ 3)dx.

(DS −

4(x−1)2 −

3 8(x−1) +

5 32ln

x−1

x+

)

16.

Z

dx x5−x2

(DS x +

1 6ln

(x−1)2

x2+x+ 1 +

1

√

3arctg 2x+

√

3 ) 17.

Z

3x2+ 8

x3+ 4x2 + 4xdx.

(DS ln|x|+ ln|x+ 2|+ 10 x+ 2) 18.

Z

2x5+ 6x3+ 1

x4+ 3x2 dx.

(DS x2−

3x −

√

3arctg x

√

19.

Z

x3+ 4x2−2x+ x4+x dx.

(DS ln|x|(x

2−x+ 1)

(x+ 1)2 +

2

√

3arctg

2x−1

√

3 ) 20.

Z

x3−3

x4+ 10x2+ 25dx.

(DS 2ln(x

2

+ 5) + 25−3x 10(x2 + 5)−

3 10

√

5arctg x

√

5)

Chı’ dˆa˜n. x4+ 10x2+ 25 = (x2+ 5)2

10.2.2 T´ıch phˆan mˆo t sˆo´ h`am vˆo ty’ do.n gia’n Mˆo.t sˆo´ t´ıch phˆan h`am vˆo ty’ thu.`o.ng g˘a.p c´o thˆe’ t´ınh du.o c b˘a`ng phu.o.ng ph´ap h˜u.u ty’ h´oa h`am du.´o.i dˆa´u t´ıch phˆan Nˆo.i dung cu’a phu.o.ng ph´ap n`ay l`a t`ım mˆo.t ph´ep biˆe´n dˆo’i du.a t´ıch phˆan d˜a cho cu’a h`am vˆo ty’ vˆe` t´ıch phˆan h`am h˜u.u ty’ Trong tiˆe´t n`ay ta tr`ınh b`ay nh˜u.ng ph´ep dˆo’i biˆe´n cho ph´ep h˜u.u ty’ h´oa dˆo´i v´o.i mˆo.t sˆo´ l´o.p h`am vˆo ty’ quan tro.ng nhˆa´t Ta quy u.´o.c k´y hiˆe.u R(x1, x2, ) hay r(x1, x2, ) l`a h`am h˜u.u

ty’ dˆo´i v´o.i mˆo˜i biˆe´nx1, x2, , xn.

I.T´ıch phˆan c´ac h`am vˆo ty’ phˆan tuyˆe´n t´ınh. T´ıch phˆan da.ng

Z

Rx,ax+b cx+d

p1

, ,ax+b cx+d

pn

dx (10.8)

trong d´o n ∈ N; p1, , pn ∈ Q; a, b, c∈ R; ad−bc 6= du.o c h˜u.u ty’

h´oa nh`o ph´ep dˆo’i biˆe´n

ax+b cx+d =t

m

o.’ dˆay m l`a mˆa˜u sˆo´ chung cu’a c´ac sˆo´ h˜u.u ty’ p1, , pn.

II T´ıch phˆan da ng

Z

R(x,

√

c´o thˆe’ h˜u.u ty’ h´oa nh`o ph´ep thˆe´ Euler: (i)

√

ax2+bx+c=±√ax±t, nˆe´u a >0;

(ii)

√

ax2+bx+c=±xt±√c, nˆe´uc >0;

(iii)√ax2+bx+c=±(x−x 1)t

√

ax2+bx+c=±(x−x 2)t

trong d´ox1 v`ax2 l`a c´ac nghiˆe.m thu c kh´ac cu’a tam th´u.c bˆa.c hai

ax2+nbx+c (Dˆa´u o.’ c´ac vˆe´ pha’i cu’a d˘a’ng th´u.c c´o thˆe’ lˆa´y theo tˆo’ ho p t`uy ´y)

III T´ıch phˆan cu’a vi phˆan nhi th´u.c D´o l`a nh˜u.ng t´ıch phˆan da.ng

Z

xm(axn+b)pdx (10.10) d´o a, b∈R,m, n, p ∈Qv`aa= 0,6 b6= 0, n6= 0,p6= 0; biˆe’u th´u.c xm(zxn+b)p

du.o c go.i l`a vi phˆan nhi th´u.c

T´ıch phˆan vi phˆan nhi th´u.c (10.10) du.a du.o c vˆe` t´ıch phˆan h`am h˜u.u ty’ ba tru.`o.ng ho p sau dˆay:

1) p l`a sˆo´ nguyˆen, 2) m+

n l`a sˆo´ nguyˆen, 3) m+

n +p l`a sˆo´ nguyˆen

D- i.nh l´y (Trebu.s´ep) T´ıch phˆan vi phˆan nhi th´u.c (10.10) biˆe’u diˆ˜ne du.o..c du.´o.i da.ng h˜u.u ha.n nh`o c´ac h`am so cˆa´p (t´u.c l`a du.a du.o c vˆe` t´ıch phˆan h`am h˜u.u ty’ hay h˜u.u ty’ h´oa du.o c) v`a chı’ ´ıt nhˆa´t mˆo.t trong ba sˆo´p, m+

n ,

m+

n +p l`a sˆo´ nguyˆen. 1) Nˆe´up l`a sˆo´ nguyˆen th`ı ph´ep h˜u.u ty’ h´oa s˜e l`a

x=tN

trong d´o N l`a mˆa˜u sˆo´ chung cu’a c´ac phˆan th´u.c m v`a n. 2) Nˆe´u m+

trong d´o M l`a mˆa˜u sˆo´ cu’ap. 3) Nˆe´u m+

n +pl`a sˆo´ nguyˆen th`ı d˘a.t a+bx−n=tM d´o M l`a mˆa˜u sˆo´ cu’ap.

C ´AC V´I DU. V´ı du 1. T´ınh

1) I1 =

Z

x+

√

x2+√6x

x(1 +√3 x) dx , 2) I2=

Z

dx

3

p

(2 +x)(2−x)5·

Gia’i. 1) T´ıch phˆan d˜a cho c´o da.ng I, d´o p1 = 1, p2 =

1 3, p3 =

1

6 Mˆa˜u sˆo´ chung cu’a p1, p2, p3 l`a m = Do d´o ta d˘a.t x = t

6

Khi d´o:

I =

Z

t6+t4+t t6(1 +t2)t

5

dt=

Z

t5+t3+ 1 +t2 dt

=

Z

t3dt+

Z

dt +t2 =

3

3

√

x2+ 6arctg√6

x+C. 2) B˘a`ng ph´ep biˆe´n dˆo’i so cˆa´p ta c´o

I2=

Z

3

r

2−x +x

dx (2−x)2 ·

D´o l`a t´ıch phˆan da.ng I Ta d˘a.t 2−x +x =t

3

v`a thu du.o c

x= 21−t

3

1 +t3, dx=−12

t2dt

T`u d´o I2 =−12

Z

t3(t3+ 1)2dt

16t6(t3+ 1)2 =−

3

Z

dt t3 =

3

3

r2 +x

2−x

2

+C. N

V´ı du 2. T´ınh c´ac t´ıch phˆan 1) I1=

Z dx

x

√

x2+x+ 1 , 2) I2 =

Z dx

(x−2)

√

−x2+ 4x−3,

3) I3=

Z

dx (x+ 1)

√

1 +x−x2 ,·

Gia’i. 1) T´ıch phˆan I1 l`a t´ıch phˆan da.ng II v`a a= >0 nˆen ta su.’

du.ng ph´ep thˆe´ Euler (i)

√

x2+x+ =x+t, x2 +x+ =x2+ 2tx+t2

x= t

2−

1 1−2t,

√

x2+x+ =x+t= −t

+t−1 1−2t dx= 2(−t

2

+t−1) (1−2t)2 dt.

T`u d´o I1 =

Z

dt

t2−1 = ln

11 +−tt

+C = ln

1 +x−

√

x2 +x+ 1

1−x+

√

x2 +x+ 1

+C.

2) Dˆo´i v´o.i t´ıch phˆan I2 (da.ng II) ta c´o

−x2+ 4x−3 =−(x−1)(x−3) v`a d´o ta su.’ du.ng ph´ep thˆe´ Euler (iii):

√

−x2+ 4x−3 =t(x−1).

Khi d´o

−(x−1)(x−3) =t2(x−1)2, −(x−3) =t2(x−1), t =

r

3−x x−1, x= t

2+ 3

t2+ 1,

√

−x2+ 4x−3 =t(x−1) = 2t

t2+ 1

v`a thu du.o..c I2 =

Z

dt

t2−1 = ln

11 +−tt+C = ln

√

x−1−√3−x

√

x−1 +√3−x

+C.

3) Dˆo´i v´o.i t´ıch phˆan I3 (da.ng III) ta c´o C = > Ta su.’ du.ng

ph´ep thˆe´ Euler (ii) v`a

√

1 +x−x2 =tx−1, 1 +x−x2

=t2x2−2tx+ 1, x= 2t+

t2+ 1 ,

√

1 +x−x2=tx−1 = t

2+t−1

t2+ 1 ,

t= +

√

1 +x−x2

x , dx =

−2(t2+t−1)

(t2+ 1)2 ·

Do d´o I3 =−2

Z

dt

t2+ 2t+ 2 =−2

Z

d(t+ 1)

1 + (t+ 1)2 =−2arctg(t+ 1) +C

=−2arctg1 +x+

√

1 +x−x2

x +C. N V´ı du 3. T´ınh c´ac t´ıch phˆan

1) I1 =

Z √

x

(1 +√3x)2dx, x>0; 2) I2 =

Z √

x4

s

1−√1

x3

dx; 3) I3 =

Z

dx x2p3

(1 +x3)5 ·

Gia’i. 1) Ta c´o

I1=

Z

x12 +x

3−2dx,

trong d´o m = 2, n =

1

3, p = −2, mˆa˜u sˆo´ chung cu’a m v`a n b˘a`ng V`ıp=−2 l`a sˆo´ nguyˆen, ta ´ap du.ng ph´ep dˆo’i biˆe´nx=t6 v`

a thu du.o c I1 =

Z

t8

(1 +t2)2dt =

Z

t4 −2t2+ 3− 4t

2+ 3

(1 +t2)2

dt =

5t

5

−4t3+ 18t−18

Z

dt +t2 −6

Z

t2

V`ı

Z

t2dt

(1 +t2)2 =−

1

Z

td 1 +t2

=− t

2(1 +t2) +

1 2arctgt nˆen cuˆo´i c`ung ta thu du.o c

I1 =

6 5x

5/6

−4x1/2+ 18x1/6+ 3x

1/6

1 +x1/3 −21arctgx 1/6

+C. 2) Ta viˆe´tI2 du.´o.i da.ng

I2 =

Z

x12 1−x−

2

1

4dx.

O’ dˆay m =

2, n=− 2, p=

1 v`a

m+

n =−1 l`a sˆo´ nguyˆen v`a ta c´o tru.`o.ng ho..p th´u hai Ta su.’ du.ng ph´ep dˆo’i biˆe´n

1− √1

x3 =t

.

Khi d´o x= (1−t4)−23, dx=

8 3(1−t

4

)−53t3dt v`a vˆa.y

I2 =

8

Z

t4

(1−t4)2dt=

2

Z

td 1−t4

=

h t

1−t4 −

Z

dt 1−t2

i

= 2t 3(1−t4) −

1

Z h 1

1−t2 +

1 +t2

i

dt = 2t

3(1−t4) −

1 6ln

1 +1−tt−

3arctgt+C, d´o t= 1−x−3/21/4.

3) Ta viˆe´tI3 du.´o.i da.ng

I3 =

Z

x−2(1 +x3)−53dx.

O’ dˆay m =−2, n = 3, p=−5

3 v`a

m+

n +p=−2 l`a sˆo´ nguyˆen Do vˆa.y ta c´o tru.`o.ng ho p th´u ba Ta thu c hiˆe.n ph´ep dˆo’i biˆe´n

T`u d´o

x3 =

t3−1, +x

= t

3

t3−1, x= (t −

1)−13

dx=−t2(t3−1)−43dt, x−2 = (t3−1)

3.

Do vˆa.y I3 =−

Z

(t3−1)2/3

t3

t3−1

−5/3

t2(t3−1)−43dt=

Z 1−t3

t3 dt

=

Z

t−3dt−

Z

dt = t −2

−2−t+C =C−

1 + 2t3

2t3

=C− + 3x

3

2xp3

B `AI T ˆA P

T´ınh c´ac t´ıch phˆan (1-15) 1.

Z

dx

√

2x−1−√3

2x−1 (DS u3+3

2u

2

+ 3u+ ln|u−1|, u6 = 2x−1) 2.

Z xdx

(3x−1)√3x−1

(DS

3x−2

√

3x−1) 3.

Z r

1−x +x

dx x

(DS 1−

√

1−x2

x −arc sinx) 4.

Z

3

r

x+ x−1

dx x+ (DS −1

2ln

(1−t)2

1 +t+t2 +

√

3arctg2t√+

3 , t=

3

r

x+ x−1) 5.

Z √

x+ 1−√x−1

√

x+ +√x−1dx. (DS

2(x

2−

x

√

x2−1 + ln|x+√x2−1|)

6.

Z

xdx

√

x+ 1−√3

x+ (DS

h1

9u

9

+1 8u

8

+1 7u

7

+1 6u

6

+1 5u

5

+ 4u

4i

7.

Z

(x−2)

r

1 +x 1−xdx.

(DS 1−1

2x

√

1−x2−

2arc sinx) 8.

Z

3

r

x+ x−1

dx (x−1)3

(DS 16

3

rx+ 1

x−1

4

−

28

3

rx+ 1

x−1

3 ) 9. Z dx p

(x−1)3(x−2) (DS

r

x−2 x−1)

Chı’ dˆa˜n. Viˆe´t p(x−1)3(x−2) = (x− 1)(x −2)

r

x−1 x−2, d˘a.t t=

r

x−1 x−2 10.

Z

dx

3

p

(x−1)2(x+ 1)

(DS 2ln

u2+u+ 1

u2−2u+ 1 −

√

3arctg2u√+

3 , u

3

= x+ x−1) 11.

Z

dx

3

p

(x+ 1)2(x−1)4 (DS

3

3

r

1 +x x−1) 12.

Z

dx

4

p

(x−1)3(x+ 2)5 (DS

4

4

r

x−1 x+ 2) 13.

Z

dx

3

p

(x−1)7(x+ 1)2 (DS

3 16

3x−5 x−1

3

r

x+ x−1) 14.

Z

dx

6

p

(x−7)7(x−5)5 (DS −3

6

r

x−5 x−7) 15.

Z

dx

n

p

(x−a)n+1(x−b)n−1, a 6=b. (DS

n b−a

n

r

16.

Z √

x+ 1−√x−1

√

x+ +√x−1dx. (DS x

2

3 −

x√x2−1

2 +

1

2ln|x+

√

x2−1|)

Su.’ du.ng c´ac ph´ep thˆe´ Euler dˆe’ t´ınh c´ac t´ıch phˆan sau dˆay (17-22) 17.

Z

dx x

√

x2+x+ 1 (DS ln

1 +x−

√

x2+x+ 1

1−x+

√

x2+x+ 1

) 18. Z dx

(x−2)√−x2+ 4x−3 (DS ln

√

x−1−√3−x

√

x−1 +√3−x

) 19. Z dx

(x+ 1)√1 +x−x2 (DS.−2arctg

1 +x+

√

1 +x−x2

x )

20.

Z

dx

(x−1)√x2+x+ 1

(DS

√

3 ln

x−1

3 + 3x+ 2p3(x2 +x+ 1)

)

21.

Z

(x−1)dx

(x2+ 2x)√x2+ 2x (DS

1 + 2x

√

x2+ 2x)

22.

Z

5x+

√

x2+ 2x+ 5dx.

(DS

√

x2+ 2x+ 5−lnx+ +

√

x2+ 2x+ 5)

Chı’ dˆa˜n. C´o thˆe’ dˆo’i biˆe´n t= 2(x

2+ 2x+ 5)0 =x+ 1. T´ınh c´ac t´ıch phˆan cu’a vi phˆan nhi th´u.c

23.

Z

x−13(1−x1/6)−1dx. (DS 6x

6 + 3x

1

3 + 2x

1

2 + lnx

6 −1)

24.

Z

x−23(1 +x

3)−3dx. (DS −3

2(1 +x

1

3)−2)

25.

Z

x−12(1 +x

4)−10dx. (DS

9(1 +x

1

4)−9−

2(1 +x

1

26.

Z

x

p

1 +

√

x2

dx. (DS

t5

5 − 2t3

3 +t

, t=

√

1 +x2/3)

27.

Z

x3(1 + 2x2)−23dx. (DS x

2+ 1

2

√

2x2+ 1)

28.

Z

dx

x4√1 +x2 (DS

1 3x

−3(2x2−1)√x2+ 1)

29.

Z

dx

x2(1 +x3)5/3 (DS −

1 8x

−1(3x+ 4)(2 +x3)−23) 30.

Z

dx

√

x3p3 1 +√4

x3

(DS −23

q

(x−3

4 + 1)2)

31.

Z

dx

3

√

x2(√3 x+ 1)3 (DS −

3 2(√3x+ 1)2)

32.

Z √3x

p

3

√

x+ 1dx. (DS u7 − 5u

+u3−u2, u2= √3

x+

)

33.

Z

dx x6√x2−1

(DS u

5

5 − 2u3

3 +u, u=

√

1−x−2)

34.

Z

dx x√3

1 +x5

(DS 10 ln

u

2 −2u+ 1

u2+u+ 1

+ √ arctg

2u+

√

3 , u

3

= +x5)

35.

Z

x7

√

1 +x2dx.

(DS u 9 − 3u7 + 3u5 − u3

3 , u

2

36.

Z

dx

3

√

1 +x3

(DS 6ln

u

2+u+ 1

u2−2u+ 1

− √1

3arctg 2u+

√

3 , u

3

= +x−3)

37.

Z

dx

4

√

1 +x4

(DS 4ln

uu+ 1−1

−

2arctgu, u

4

= +x−4) 38.

Z

3

√

x−x3dx.

(DS u 2(u3+ 1) −

1 12ln

u2+ 2u+ 1

u2−u+ 1 −

1

√

3arctg

2u−1

√

3 , u

3

=x−2−1) 10.2.3 T´ıch phˆan c´ac h`am lu.o..ng gi´ac

I T´ıch phˆan da.ng

Z

R(sinx,cosx)dx (10.11) d´o R(u, v) l`a h`am h˜u.u ty’ cu’a c´ac biˆe´nu b`a v luˆon luˆon c´o thˆe’ h˜u.u ty’ h´oa du.o c nh`o ph´ep dˆo’i biˆe´n t = tgx

2, x∈(−π, π) T`u d´o sinx= 2t

1 +t2, cosx=

1−t2

1 +t2, dx=

2dt +t2 ·

Nhu.o c diˆe’m cu’a ph´ep h˜u.u ty’ h´oa n`ay l`a n´o thu.`o.ng du.a dˆe´n nh˜u.ng t´ınh to´an rˆa´t ph´u.c ta.p.

V`ı vˆa.y, nhiˆe`u tru.`o.ng ho p ph´ep h˜u.u ty’ h´oa c´o thˆe’ thu c hiˆe.n du.o c nh`o nh˜u.ng ph´ep dˆo’i biˆe´n kh´ac

II Nˆe´u R(−sinx,cosx) = −R(sinx,cosx) th`ı su.’ du.ng ph´ep dˆo’i biˆe´n

v`a l´uc d´o

dx =−√ dt

1−t2

III Nˆe´u R(sinx,−cosx) = −R(sinx,cosx) th`ı su.’ du.ng ph´ep dˆo’i biˆe´n

t= sinx, dx = √ dt

1−t2, x∈

− π

2, π

.

IV Nˆe´u R(−sinx,−cosx) = R(sinx,cosx) th`ı ph´ep h˜u.u ty’ h´oa s˜e l`a t= tgx, x∈

− π

2, π

: sinx = √ t

1 +t2, cosx=

1

√

1 +t2, x= arctgt, dx=

dt +t2·

V Tru.`o.ng ho p riˆeng cu’a t´ıch phˆan da.ng (10.11) l`a t´ıch phˆan

Z

sinmxcosnxdx, m, n∈Z (10.12) (i) Nˆe´u sˆo´m le’ th`ı d˘a.tt = cosx, nˆe´un le’ th`ı d˘a.t sinx=t.

(ii) Nˆe´umv`anl`a nh˜u.ng sˆo´ ch˘a˜n khˆong ˆam th`ı tˆo´t ho.n hˆe´t l`a thay sin2x v`a cos2x theo c´ac cˆong th´u.c

sin2x=

2(1−cos 2x), cos

2

x=

2(1 + cos 2x)

(iii) Nˆe´um v`a n ch˘a˜n, d´o c´o mˆo.t sˆo´ ˆam th`ı ph´ep dˆo’i biˆe´n s˜e l`a tgx =t hay cotgx=t.

(iv) Nˆe´u m+n = −2k, k ∈ N th`ı viˆe´t biˆe’u th´u.c du.´o.i dˆa´u t´ıch phˆan bo.’ i da.ng phˆan th´u.c v`a t´ach cos2x (ho˘

a.c sin2x) kho’i mˆa˜u sˆo´ Biˆe’u th´u.c dx

cos2x (ho˘a.c

dx

sin2x) du.o c thay bo.’i d(tgx) (ho˘a.c d(cotgx)) v`a ´ap du.ng ph´ep dˆo’i biˆe´n t = tgx (ho˘a.c t= cotgx)

VI T´ıch phˆan da.ng

Z

B˘a`ng ph´ep dˆo’i biˆe´n sin2x=t ta thu du.o..c I =

2

Z

tα−21(1−t)

β−1

2 dt

v`a b`ai to´an du.o c quy vˆe` t´ıch phˆan cu’a vi phˆan nhi th´u.c. C ´AC V´I DU.

V´ı du 1. T´ınh t´ıch phˆan I =

Z

dx

3 sinx+ cosx+

Gia’i. D˘a.t t = tgx

2, x∈(−π, π) Khi d´o I =

Z dt

t2 + 6t+ 9 =

Z

(t+ 3)−2dt =−

t+ +C =− + tgx

2

+C. N

V´ı du 2. T´ınh

J =

Z

dx

(3 + cos 5x) sin 5x

Gia’i. D˘a.t 5x=t Ta thu du.o..c J =

5

Z

dt (3 + cost) sint

v`a (tru.`o.ng ho p II) d´o b˘a`ng c´ach d˘a.t ph´ep dˆo’i biˆe´n z = cost ta c´o J =

5

Z

dz

(z+ 3)(z2−1) =

1

Z h A

z−1+ B z−1+

C z+

i

dz =

5

Z h 1

8(z−1) − 4(z+ 1) +

1 8(z+ 3)

i

dz =

5

h1

8ln|z−1| −

4ln|z+ 1|+

8ln|z+ 3|

i

+C =

40 ln

(z−(z1)(z+ 1)+ 3)2

+C

= 40 ln

cos

2x+ cos 5x−3

(cos 5x+ 1)2

V´ı du 3. T´ınh

J =

Z

2 sinx+ cosx sin2xcosx+ cos3xdx

Gia’i. H`am du.´o.i dˆa´u t´ıch phˆan c´o t´ınh chˆa´t l`a R(−sinx,−cosx) = R(sinx,cosx). Do d´o ta su.’ du.ng ph´ep dˆo’i biˆe´n t = tgx, x∈

− π

2, π

Chia tu.’ sˆo´ v`a mˆa˜u sˆo´ cu’a biˆe’u th´u.c du.´o.i dˆa´u t´ıch phˆan cho cos3x ta c´o

J =

Z

2tgx+

tg2x+ 9d(tgx) =

Z

2t+ t2+ 9dt

= ln(t2+ 9) + arctgt

+C = ln(tg2x+ 9) + arctg

tgx

3

+C. N

V´ı du 4. T´ınh

J =

Z

dx sin6x+ cos6x

Gia’i. Ap du.ng cˆong th´u´ c

cos2x=

2(1 + cos 2x), sin

2

x=

2(1−sin 2x) ta thu du.o c

cos6x+ sin6x=

4(1 + cos

2

2x) D˘a.t t = tg2x, ta t`ım du.o c

J =

Z

4dx

1 + cos22x =

Z

dt t2+ 4

= arctgt

2+C = arctg tg2x

V´ı du 5. T´ınh

J =

Z

sin32 xcos

2 xdx.

Gia’i. D˘a.t z = sin2x ta thu du.o c J =

2

Z

z1/4(1−z)−14dx.

D´o l`a t´ıch phˆan cu’a vi phˆan nhi th´u.c v`a m+

n +p= 4+

1 −

1 = Do vˆa.y ta thu..c hiˆe.n ph´ep dˆo’i biˆe´n

1

z −1 =t

4

, −dz

z2 = 4t

dt, z2 = (t4+ 1)2

v`a d´o

J =−2

Z

t2

(t4+ 1)2dt.

D˘a.t t =

y ta thu du.o c J =

Z y4

(1 +y4)2dy.

Thu c hiˆe.n ph´ep t´ıch phˆan t`u.ng phˆa`n b˘a`ng c´ach d˘a.t u=y, dv= y

3

(1 +y4)2dy⇒du=dy, v=−

1 4(1 +y2)

ta thu du.o c

J = 2h− y

4(1 +y4) +

1

Z

dy +y4

i

=− y

2(1 +y4)+

Dˆe’ t´ınh J1 ta biˆe’u diˆ˜n tu.e ’ sˆo´ cu’a biˆe’u th´u.c du.´o.i dˆa´u t´ıch phˆan

nhu sau:

1 =

(y2+ 1)−(y2−1) v`a d´o

J1 =

1

Z

y2+ 1

y4+ 1dy−

1

Z

y2−1

y4+ 1dy

=

Z +

y2

dy y2+

y2

−

2

Z 1−

y2

dy y2+

y2

=

Z d

y+ y

y−

y + −1 Z d

y+ y

y+1 y −2 = √ 2arctg y−

y √ − √ 2ln

y+ y −

√

2 yb+1

y +

√

2

+C.

Cuˆo´i c`ung ta thu du.o..c

J =− y

2(1 +y4) +

1

√

2arctg y−

4 √ − √ 2ln y+ y − √ y+

y +

√

2

+C

trong d´o

y=

t , t =

4

r

1

z −1, z = sin

2

x. N

B `AI T ˆA P

1.

Z

sin3xdx. (DS.−cosx+ cos

3

x ) 2.

Z

cos4xdx. (DS 3x + sin 2x + sin 4x 32 ) 3. Z

sin5xdx. (DS 3cos

3x− cos 5x

5 −cosx) 4.

Z

cos7xdx. (DS sinx−sin3x+3 sin

5

x

5 −

sin7x ) 5.

Z

cos2xsin2xdx. (DS x −

sin 4x 32 ) 6.

Z

sin3xcos2xdx. (DS cos

5x

5 −

cos3x

3 ) 7.

Z

cos3xsin5xdx. (DS sin

6

x

6 −

sin8x ) 8.

Z

dx

sin 2x (DS

2ln|tgx|) 9.

Z

dx cosx

3

(DS ln

tgπ

4 + x ) 10. Z

sinx+ cosx

sin 2x dx. (DS h ln tgx + ln tgπ

4 + x i ) 11. Z

sin2x

cos6xdx. (DS

tg5x

5 + tg3x

3 )

Chı’ dˆa˜n. D˘a.t t= tgx 12.

Z

sin 3xcosxdx. (DS −1

8(cos 4x+ cos 2x)) 13.

Z

sinx 3cos

2x

3 dx. (DS 2cos

x −

1 2cosx) 14.

Z

cos3x

sin2xdx. (DS −

sinx −sinx) 15.

Z

sin3x

cos2xdx. (DS

1

16.

Z

cos3x

sin5xdx. (DS −

cotg4x ) 17.

Z

sin5x

cos3xdx. (DS

1

2 cos2x + ln|cosx| −

cos2x ) 18.

Z

tg5xdx. (DS tg

4x

4 − tg2x

2 −ln|cosx|)

Trong c´ac b`ai to´an sau dˆay h˜ay ´ap du.ng ph´ep dˆo’i biˆe´n t= tgx

2, sinx= 2t

1 +t2, cosx=

1−t2

1 +t2, x= 2arctgt, dx =

2dt +t2

19.

Z

dx

3 + cosx (DS 4ln

2 + tgx 2−tgx ) 20. Z dx

sinx+ cosx (DS

√

2ln

tgx

2 + π ) 21. Z

3 sinx+ cosx sinx+ cosxdx.

(DS

13(12x−5 ln|2tgx+ 3| −5 ln|cosx|) 22.

Z

dx

1 + sinx+ cosx (DS ln

1 + tgx ) 23. Z dx

(2−sinx)(3−sinx) (DS √2

3arctg 2tgx

2 −1

√

3

−√1

2arctg 3tgx

2 −1

√

2 ) T´ınh c´ac t´ıch phˆan da.ng

Z

sinmxcosnxdx, m, n∈N 24.

Z

sin3xcos5xdx. (DS 8cos

8x−1

6cos

25.

Z

sin2xcos4xdx. (DS 16 x−

1

4sin 4x+ 3sin

2

2x) 26.

Z

sin4xcos6xdx.

(DS

211sin 8x−

1

28 sin 4x+

1 5·26 sin

5

2x+ 28x)

27.

Z

sin4xcos2xdx. (DS x 16 −

sin 4x 64 −

sin22x 48 ) 28.

Z

sin4xcos5xdx. (DS 5sin

5

x−

7sin

7

x+1 9sin

9

x) 29.

Z

sin6xcos3xdx. (DS 7sin

7

x−

9sin

9

x) T´ınh c´ac t´ıch phˆan da.ng

Z

sinαxcosβxdx, α, β ∈Q 30.

Z

sin3x

cosx√3cosxdx. (DS

3 5cosx

3

√

cos2x+

3

√

cosx)

Chı’ dˆa˜n. D˘a.t t= cosx. 31.

Z

dx

3

√

sin11xcosx (DS

− 3(1 + 4tg

2x)

8tg2x·p3

tg2x)

Chı’ dˆa˜n. D˘a.t t= tgx 32.

Z

sin3x

3

√

cos2xdx. (DS

3

√

cosx 7cos

2x−1)

33.

Z

3

√

cos2xsin3

xdx. (DS −3

5cos

5/3x+

11cos 11 x) 34. Z dx √

sin3xcos5x (DS

4

√

tgx) 35.

Z

sin3x

5

√

cosxdx. (DS 14cos

14

5 x−

4cos

4

T´ıch phˆan x´ac di.nh Riemann

11.1 H`am kha’ t´ıch Riemann v`a t´ıch phˆan x´ac di.nh 58

11.1.1 D- i.nh ngh˜ıa 58 11.1.2 D- iˆe`u kiˆe.n dˆe’ h`am kha’ t´ıch 59 11.1.3 C´ac t´ınh chˆa´t co ba’n cu’a t´ıch phˆan x´ac di.nh 59

11.2 Phu.o.ng ph´ap t´ınh t´ıch phˆan x´ac d i.nh 61 11.3 Mˆo.t sˆo´ ´u.ng du ng cu’ a t´ıch phˆan x´ac d i.nh 78

11.3.1 Diˆe.n t´ıch h`ınh ph˘a’ng v`a thˆe’ t´ıch vˆa.t thˆe’ 78 11.3.2 T´ınh dˆo d`ai cung v`a diˆe.n t´ıch m˘a.t tr`on xoay 89

11.4 T´ıch phˆan suy rˆo.ng 98

11.1 H`am kha’ t´ıch Riemann v`a t´ıch phˆan x´ac di.nh

11.1.1 D- i.nh ngh˜ıa

Gia’ su.’ h`am f(x) x´ac di.nh v`a bi ch˘a.n trˆen doa.n [a, b] Tˆa.p ho p h˜u.u ha.n diˆe’mxk nk=0:

a=x0 < x1 < x2 <· · ·< xn−1 < xn=b

du.o c go.i l`a ph´ep phˆan hoa.ch doa.n [a, b] v`a du.o..c k´y hiˆe.u l`a T[a, b] hay do.n gia’n l`aT

D- i.nh ngh˜ıa 11.1.1. Gia’ su.’ [a, b] ⊂ R, T[a, b] = {a = x0 < x1 <

· · ·< xn =b}l`a ph´ep phˆan hoa.ch doa.n [a, b] Trˆen mˆo˜i doa.n [xj−1, xj],

j = 1, , n ta cho.n mˆo.t c´ach t`uy ´y diˆe’m ξj v`a lˆa.p tˆo’ng S(f, T, ξ) =

n

X

j=1

f(ξj)∆xj, ∆xj =xj −xj−1

go.i l`a tˆo’ng t´ıch phˆan (Riemann) cu’a h`am f(x) theo doa.n [a, b] tu.o.ng ´

u.ng v´o.i ph´ep phˆan hoa.ch T v`a c´ach cho.n diˆe’m ξj,j = 1, n Nˆe´u gi´o.i ha.n

lim

d(T)→0S(f, T, ξ) = limd(T)→0

n

X

j=1

f(ξj)∆xj (11.1)

tˆ` n ta.i h˜u.u ha.n khˆong phu thuˆo.c v`ao ph´ep phˆan hoa.cho T v`a c´ach cho.n c´ac diˆe’m ξj, j = 1, n th`ı gi´o.i ha.n d´o du.o c go.i l`a t´ıch phˆan x´ac di.nh cu’a h`am f(x).

11.1.2 D- iˆ`u kiˆe e.n dˆe’ h`am kha’ t´ıch

D- i.nh l´y 11.1.1. Nˆe´u h`amf(x)liˆen tu c trˆen doa n[a, b]th`ıf ∈ R[a, b].

Hˆe qua’. Mo i h`am so cˆa´p dˆ`u kha’ t´ıch trˆen doa.n bˆa´t k`y n˘a`m tro.ne trong tˆa p ho p x´ac di.nh cu’a n´o.

D- i.nh l´y 11.1.2. Gia’ su.’ f : [a, b] →R l`a h`am bi ch˘a.n v`a E ⊂ [a, b]

l`a tˆa p ho p c´ac diˆe’m gi´an doa.n cu’a n´o H`am f(x) kha’ t´ıch Riemann trˆen doa n [a, b] khi v`a chı’ tˆa p ho..p E c´o dˆo - khˆong, t´u.c l`a E

tho’a m˜an diˆ`u kiˆe.n:e ∀ε >0, tˆ` n ta.i hˆe dˆe´m du.o c (hay h˜u.u ha.n) c´aco khoa’ng (ai, bi) sao cho

E ⊂

∞

[

i=1

(ai, bi), ∞

X

i=1

(bi−ai) = lim N→∞

N

X

i=1

(bi−ai)< ε.

Nˆe´u c´ac diˆ`u kiˆe.n cu’a di.nh l´y 11.1.2 (go.i l`a tiˆeu chuˆa’n kha’ t´ıche Lo.be (Lebesgue)) du.o c tho’a m˜an th`ı gi´a tri cu’a t´ıch phˆan

b

Z

a

f(x)dx khˆong phu thuˆo.c v`ao gi´a tri cu’a h`amf(x) ta.i c´ac diˆe’m gi´an doa.n v`a ta.i c´ac diˆe’m d´o h`am f(x) du.o..c bˆo’ sung mˆo.t c´ach t`uy ´y nhu.ng pha’i ba’o to`an t´ınh bi ch˘a.n cu’a h`am trˆen [a, b].

11.1.3 C´ac t´ınh chˆa´t co ba’n cu’a t´ıch phˆan x´ac di.nh

1) a

Z

a

f(x)dx=

2) b

Z

a

f(x)dx=−

a

Z

b

f(x)dx.

4) Nˆe´uf ∈ R[a, b] th`ı|f(x)| ∈ R[a, b] v`a

b

Z

a

f(x)dx

6

b

Z

a

|f(x)|dx, a < b. 5) Nˆe´uf, g ∈ R[a, b] th`ıf(x)g(x)∈ R[a, b]

6) Nˆe´uf g ∈ D[a, b] v`a ]c, d]⊂[a, b] th`ıf(x)g(x)∈ R[c, d]

7) Nˆe´u f ∈ R[a, c], f ∈ R[c, b] th`ıf ∈ R[a, b], d´o diˆe’m c c´o thˆe’ s˘a´p xˆe´p t`uy ´y so v´o.i c´ac diˆe’m a v`a b.

Trong c´ac t´ınh chˆa´t sau dˆay ta luˆon luˆon xem a < b. 8) Nˆe´uf ∈ R[a, b] v`a f >0 th`ı

b

Z

a

f(x)dx>0 9) Nˆe´uf, g ∈ R[a, b] v`af(x)>g(x)∀x∈[a, b] th`ı

b

Z

a

f(x)dx>

b

Z

a

g(x)dx.

10) Nˆe´uf ∈ C[a, b],f(x)> 0,f(x) 6≡0 trˆen [a, b] th`ı∃K >0 cho

b

Z

a

f(x)dx>K.

11) Nˆe´uf, g∈ R[a, b],g(x)>0 trˆen [a, b] M = sup

[a,b]

f(x), m= inf

[a,b]f(x)

th`ı

m b

Z

a

g(x)dx6

b

Z

a

f(x)g(x)dx≤M b

Z

a

11.2 Phu.o.ng ph´ap t´ınh t´ıch phˆan x´ac di.nh

Gia’ su.’ h`am f(x) kha’ t´ıch trˆen doa.n [a, b] H`am

F(x) = x

Z

a

f(x)dt, a 6x 6b

du.o c go.i l`a t´ıch phˆan v´o.i cˆa.n trˆen biˆe´n thiˆen

D- i.nh l´y 11.2.1. H`am f(x) liˆen tu c trˆen doa n [a, b]l`a c´o nguyˆen h`am trˆen doa n d´o Mˆo t c´ac nguyˆen h`am cu’a h`am f(x) l`a h`am

F(x) = x

Z

a

f(t)dt. (11.2)

T´ıch phˆan v´o.i cˆa.n trˆen biˆe´n thiˆen du.o c x´ac di.nh dˆo´i v´o.i mo.i h`am f(x) kha’ t´ıch trˆen [a, b] Tuy nhiˆen, dˆe’ h`amF(x) da.ng (11.2) l`a nguyˆen h`am cu’a f(x) diˆ`u cˆo´t yˆe´u l`ae f(x) pha’i liˆen tu.c.

Sau dˆay l`a di.nh ngh˜ıa mo.’ rˆo.ng vˆe` nguyˆen h`am

D- i.nh ngh˜ıa 11.2.1. H`am F(x) du.o c go.i l`a nguyˆen h`am cu’a h`am f(x) trˆen doa.n [a, b] nˆe´u

1) F(x) liˆen tu.c trˆen [a, b].

2) F0(x) =f(x) ta.i c´ac diˆe’m liˆen tu.c cu’af(x)

Nhˆa n x´et. H`am liˆen tu.c trˆen doa.n [a, b] l`a tru.`o.ng ho..p riˆeng cu’a h`am liˆen tu.c t`u.ng doa.n Do d´o dˆo´i v´o.i h`am liˆen tu.c di.nh ngh˜ıa 11.2.1 vˆ` nguyˆen h`am l`a tr`e ung v´o.i di.nh ngh˜ıa c˜u tru.´o.c dˆay v`ıF0(x) =f(x)

nguyˆen h`am l`a

F(x) = x

Z

a

f(t)dt.

D- i.nh l´y 11.2.3. (Newton-Leibniz) Dˆo´i v´o.i h`am liˆen tu c t`u.ng doa n trˆen [a, b] ta c´o cˆong th´u.c Newton-Leibniz:

b

Z

a

f(x)dx=F(b)−F(a) (11.3)

trong d´o F(x) l`a nguyˆen h`am cu’a f(x) trˆen [a, b] v´o.i ngh˜ıa mo.’ rˆo ng.

D- i.nh l´y 11.2.4 (Phu.o.ng ph´ap dˆo’i biˆe´n) Gia’ su.’ :

(i) f(x) x´ac di.nh v`a liˆen tu.c trˆen [a, b],

(ii) x = g(t) x´ac di.nh v`a liˆen tu.c c`ung v´o.i da.o h`am cu’a n´o trˆen doa n [α, β], d´o g(α) =a, g(β) =bv`a a6g(t)6b.

Khi d´o

b

Z

a

f(x)dx= β

Z

α

f(g(t))g0(t)dt (11.4) D- i.nh l´y 11.2.5 (Phu.o.ng ph´ap t´ıch phˆan t`u.ng phˆ` n).a Nˆe´u f(x) v`a

g(x) c´o da o h`am liˆen tu c trˆen [a, b]th`ı

b

Z

a

f(x)g0(x)dx=f(x)g(x)ba−

b

Z

a

f0(x)g(x)dx (11.5)

C ´AC V´I DU. V´ı du 1. Ch´u.ng to’ r˘a`ng trˆen doa.n [−1,1] h`am

f(x) = signx=

        

1 v´o.i x > 0,

0 v´o.i x= 0, x∈[−1,1]

a) kha’ t´ıch, b) khˆong c´o nguyˆen h`am, c) c´o nguyˆen h`am mo.’ rˆo.ng.

Gia’i. a) H`am f(x) kha’ t´ıch v`ı n´o l`a h`am liˆen tu.c t`u.ng doa.n b) Ta ch´u.ng minh h`am f(x) khˆong c´o nguyˆen h`am theo ngh˜ıa c˜u Thˆa.t vˆa.y mo.i h`am da.ng

F(x) =

  

−x+C1 x <0

x+C2 x>0

dˆ`u c´o da.o h`am b˘a`ng signe x ∀x 6= 0, d´o C1 v`a C2 l`a c´ac sˆo´ t`uy

´

y Tuy nhiˆen, thˆa.m ch´ı h`am “tˆo´t nhˆa´t” sˆo´ c´ac h`am n`ay F(x) =|x|+C

(nˆe´u C1 = C2 = C) c˜ung khˆong c´o da.o h`am ta.i diˆe’m x = Do d´o

h`am signx (v`a d´o mo.i h`am liˆen tu.c t`u.ng doa.n) khˆong c´o da.o h`am trˆen khoa’ng bˆa´t k`y ch´u.a diˆe’m gi´an doa.n.

c) Trˆen doa.n [−1,1] h`am signx c´o nguyˆen h`am mo.’ rˆo.ng l`a h`am F(x) =|x|v`ı n´o liˆen tu.c trˆen doa.n [−1,1] v`aF0(x) =f(x) khix6=

N

V´ı du 2. T´ınh a

Z

0

√

a2−x2dx, a >0.

Gia’i. D˘a.t x= asint Nˆe´u t cha.y hˆe´t doa.n h0,π

i

th`ıx cha.y hˆe´t doa.n [0, a] Do d´o

a

Z

0

√

a2−x2dx=

π/2

Z

0

a2cos2tdt=a2 π/2

Z

0

1 + cos 2t dt

= a

2

2 π/2

Z

0

dt+a

2

2 π/2

Z

0

cos 2tdt= a

2π

4 · N V´ı du 3. T´ınh t´ıch phˆan

I = √

2/2

Z

0

r

Gia’i. Ta thu..c hiˆe.n ph´ep dˆo’i biˆe´n x = cost Ph´ep dˆo’i biˆe´n n`ay tho’a m˜an c´ac diˆ`u kiˆe.n sau:e

(1) x=ϕ(t) = cost liˆen tu.c∀t ∈R (2) Khit biˆe´n thiˆen trˆen doa.n

hπ

4, π

i

th`ıxcha.y hˆe´t doa.n

h

0,

√

2

i

(3) ϕ

π

4

=

√

2 , ϕ

π

2

= (4) ϕ0(t) = −sint liˆ

en tu.c ∀t∈hπ

4, π

i

Nhu vˆa.y ph´ep dˆo’i biˆe´n tho’a m˜an di.nh l´y 11.2.4 v`a d´o x= cost, dx=−sintdt,

ϕ

π

2

= 0, ϕ

π

4

=

√

2 · Nhu vˆa.y

I =

π

4

Z

−π2

cotgt

2(−sint)dt=

π

2

Z

π

4

(1 + cost)dt

=t+ sintπ/π/24 = π + 1−

√

2 ·. N V´ı du 4. T´ınh t´ıch phˆan

I = √

3/2

Z

1/2

dx x

√

1−x2·

Gia’i. Ta thu c hiˆe.n ph´ep dˆo’i biˆe´n

x= sint⇒dx= costdt v`a biˆe’u th´u.c du.´o.i dˆa´u t´ıch phˆan c´o da.ng

costdt sint

√

cos2t =

    

dt

sint nˆe´u cost > 0,

− dt

C´ac cˆa.n α v`a β cu’a t´ıch phˆan theot du.o c x´ac di.nh bo.’i

2 = sint ⇒α= π 6,

√

3

2 = sint ⇒β = π 3· (Ta c˜ung c´o thˆe’ lˆa´y α1 =

5π

6 v`a β1 = 2π

3 ) Trong ca’ hai tru.`o.ng ho p biˆe´n x= sint dˆ`u cha.y hˆe´t doa.n [e a, b] =

h1

2,

√

3

i

Ta s˜e thˆa´y kˆe´t qua’ t´ıch phˆan l`a nhu nhau Thˆa.t vˆa.y tru.`o.ng ho p th´u nhˆa´t ta c´o cost >0 v`a

I = π/3

Z

π/6

dt

sint = ln tg t

π/3

Z

π/6

= ln2 +

√

3 · Trong tru.`o.ng ho p th´u hait∈h5π

6 , 2π

3

i

ta c´o cost <0 v`a

I =−

2π/3

Z

5π/6

dt

sint =−ln tg t

2π/3 5π/6

= ln2 +

√

3

3 · N

V´ı du 5. T´ınh t´ıch phˆan

I = π/3

Z

0

xsinx cos2xdx.

Gia’i. Ta t´ınh b˘a`ng phu.o.ng ph´ap t´ıch phˆan t`u.ng phˆ` n.a D˘a.t

u=x⇒du=dx, dv = sinxdx

cos2x ⇒v=

Do d´o

I =x·

cosx π/3 − π/3 Z dx cosx =

π cosπ

3

−ln tgx + π π/3 = 2π

3 −ln tg

π

6 + π

+ ln tgπ =

2π

3 −ln tg 5π 12 · N V´ı du 6. T´ınh t´ıch phˆan

I =

1

Z

0

x2(1−x)3dx.

Gia’i. Ta d˘a.t

u=x2, dv= (1−x)3dx⇒

du= 2xdx, v=−(1−x)

4

4 ·

Do d´o

I =−x2(1−x)

4 0+ Z

2x(1−x)

4

4 dx

| {z }

I1

= +I1.

T´ınh I1 T´ıch phˆan t`u.ng phˆ` na I1 ta c´o

I1 =

1

1

Z

0

x(1−x)4dx=−1

2x

(1−x)5

5 0+ Z

(1−x)5 dx = 0−

10

(1−x)6

6 =

60 ⇒I = 60 · N

V´ı du 7. Ap du.ng cˆong th´u.c Newton-Leibnitz dˆe’ t´ınh t´ıch phˆan´

1) I1 = 100π

Z

0

√

1−cos 2xdx, 2) I2=

Z

0

Gia’i. Ta c´o

√

1−cos 2x=

√

2|sinx| Do d´o

100π

Z

0

√

1−cos 2xdx=

√

2

100π

Z

0

|sinx|dx =

√

2h π

Z

0

sinxdx−

2π

Z

π

sinxdx+

3π

Z

2π

sinxdx− . +· · ·+

100Z π

99π

sinxdx

i

=− √

2[2 + +· · ·+ 2] = 200

√

2

2) Thu c hiˆe.n ph´ep dˆo’i biˆe´n t=e−x, sau d´o ´ap du.ng phu.o.ng ph´ap t´ıch phˆan t`u.ng phˆ` n Ta c´oa

Z

exarc sin(e−x)dx=−

Z

arc sint t2 dt

=

tarc sint−

Z

dt t

√

1−t2

=

tarc sint+I1.

I1 =−

Z

dt t√1−t2 =

Z d1

t

r1

t

2

−1

= ln1 t +

r

1 t2 −1

+C.

Do d´o

Z

exarc sine−xdx= arc sint t + ln

1

t +

r1

t2 −1

+C =exarc sine−x+ ln(ex+

√

e2x−1) +C Nguyˆen h`am v`u.a thu du.o c c´o gi´o.i ha.n h˜u.u ha.n ta.i diˆe’m x = d´o theo cˆong th´u.c (11.3) ta c´o

1

Z

0

exarc sine−xdx=earc sine−1 −π

2 + ln(e+

√

V´ı du 8. T´ınh t´ıch phˆan Dirichlet π/2

Z

0

sin(2n−1)x

sinx dx, n∈N

Gia’i. Ta c´o cˆong th´u.c +

n−1

X

k=1

cos 2kx= sin(2n−1)x sinx · T`u d´o v`a lu.u ´y r˘a`ng

π/2

Z

0

cos 2kxdx= 0, k= 1,2, , n−1 ta c´o π/2

Z

0

sin(2n−1)x sinx dx=

π 2· N B `AI T ˆA P

T´ınh c´ac t´ıch phˆan sau dˆay b˘a`ng phu.o.ng ph´ap dˆo’i biˆe´n (1-14) 1.

5

Z

0

xdx

√

1 + 3x (DS 4)

2.

ln

Z

ln

dx

ex−e−x (DS ln3

2 )

3. √

3

Z

1

(x3+ 1)dx

x2√4−x2 (DS

7

√

3 −1) D˘a.t x= sint. 4.

π/2

Z

0

dx

2 + cosx (DS π

√

5.

1

Z

0

x2dx

(x+ 1)4 (DS

1 24)

6.

ln

Z

0

√

ex−1dx. (DS. 4−π )

7. √

7

Z

√

3

x3dx

3

p

(x2+ 1)2 (DS 3)

Chı’ dˆa˜n. D˘a.t t=x2+ 1.

8. e

Z

1

4

√

1 + lnx

x dx. (DS 0,8(2

4

√

2−1))

Chı’ dˆa˜n. D˘a.t t= + lnx.

9.

+√3

Z

−3

x2

√

9−x2dx. (DS. 81π

8 )

chı’ dˆa˜n. D˘a.t x= cost.

10.

3

Z

0

r

x

6−xdx. (DS

3(π−2)

2 )

Chı’ dˆa˜n. D˘a.t x= sin2t. 11.

π

Z

0

sin6x

2dx. (DS 5π 16)

Chı’ dˆa˜n. D˘a.t x= 2t

12. π/4

Z

0

cos72xdx (DS 35)

13. √

2/2

Z

0

r

1 +x

1−xdx. (DS π

4 + 1−

√

2 )

Chı’ dˆa˜n. D˘a.t x= cost. 14.

29

Z

3

3

p

(x−2)2

3 +p3

(x−2)2dx. (DS +

3

√

3 π)

T´ınh c´ac t´ıch phˆan sau dˆay b˘a`ng phu.o.ng ph´ap t´ıch phˆan t`u.ng phˆ` n (15-32).a

15.

1

Z

0

x3arctgxdx (DS 6)

16. e

Z

1/e

|lnx|dx. (DS 2(1−1/e)) 17.

π

Z

0

exsinxdx. (DS 2(e

π + 1))

18.

1

Z

0

x3e2xdx. (DS e

2+ 3

8 )

19.

1

Z

0

arc sinx

√

1 +xdx. (DS π

√

2−4) 20.

π/4

Z

0

ln(1 + tgx)dx (DS πln )

21. π/b

Z

0

eaxsinbxdx. (DS b a2+b2 e

πa b + 1)

22.

1

Z

0

e−xln(ex+ 1)dx (DS −1 +e

23. π/2

Z

0

sin 2x·arctg(sinx)dx. (DS π −1) 24.

2

Z

1

sin(lnx)dx. (DS sin(ln 2)−cos(ln 2) + 2)

25. π

Z

0

x3sinxdx. (DS π3−6π)

26.

2

Z

1

xlog2xdx. (DS 2−

4 ln 2)

27. a

√

7

Z

0

x3

3

√

a2+x2dx. (DS

141a3√3 a

20 )

28. a

Z

0

√

a2−x2dx. (DS. πa

4 )

29. π/2

Z

π/6

x+ sinx

1 + cosxdx. (DS π

6(1 +

√

3))

30. π/2

Z

0

sinmxcos(m+ 2)xdx (DS −

cos mπ m+ )

31. π/2

Z

0

cosmxcos(m+ 2)xdx (DS 0)

32. π/2

Z

0

cosxcos 2nxdx (DS π 4n(−1)

33. T´ınh

2

Z

0

f(x)dx, d´o

f(x) =

  

x2 khi 06x61

2−x 16x62

b˘a`ng hai phu.o.ng ph´ap; a) su.’ du.ng nguyˆen h`am cu’a f(x) trˆen doa.n [0,2]; b) chia doa.n [0,2] th`anh hai doa.n [0,1] v`a [1,2] (DS

6) 34. Ch´u.ng minh r˘a`ng nˆe´u f(x) liˆen tu.c trˆen doa.n [−`, `] th`ı

(i) `

Z

−`

f(x)dx = `

Z

0

f(x)dx f(x) l`a h`am ch˘a˜n;

(ii) `

Z

−`

f(x)dx = f(x) l`a h`am le’

35. Ch´u.ng minh r˘a`ng ∀m, n ∈ Z c´ac d˘a’ng th´u.c sau dˆay du.o c tho’a m˜an:

(i) π

Z

−π

sinmxcosnxdx=

(ii) π

Z

−π

cosmxcosnxdx= 0, m 6=n. (iii)

π

Z

−π

sinmxsinnxdx= 0, m 6=n. 36. Ch´u.ng minh d˘a’ng th´u.c

b

Z

a

f(x)dx= b

Z

a

f(a+b−x)dx.

37. Ch´u.ng minh d˘a’ng th´u.c π/2

Z

0

f(cosx)dx= π/2

Z

0

f(sinx)dx.

Chı’ dˆa˜n. D˘a.t t= π −x.

38. Ch´u.ng minh r˘a`ng nˆe´u f(x) liˆen tu.c khi x>0 th`ı a

Z

0

x3f(x2)dx=

a2

Z

0

xf(x)dx

39. Ch´u.ng minh r˘a`ng nˆe´u f(t) l`a h`am le’ th`ı x

Z

a

f(t)dt l`a h`am ch˘a˜n, t´u.c l`a

−x

Z

a

f(t)dt = x

Z

a

f(t)dt

Chı’ dˆa˜n. D˘a.t t=−xv`a biˆe’u diˆe˜n −x

Z

−a

f(t)dt= a

Z

−a +

−x

Z

a

v`a su.’ du.ng t´ınh ch˘a˜n le’ cu’a h`am f.

T´ınh c´ac t´ıch phˆan sau dˆay (40-65) b˘a`ng c´ach ´ap du.ng cˆong th´u.c Newton-Leibnitz

40.

5

Z

0

xdx

√

1 + 3x (DS 4)

41.

ln

Z

ln

dx

ex−e−x (DS

42. √

3

Z

0

(x3+ 1)dx

x2√4−x2 (DS

7

√

3

−1) 43.

π/2

Z

0

dx

2 + cosx (DS π

√

3)

44.

ln

Z

0

√

ex−1dx. (DS. 4−π )

45. √

7

Z

√

3

x3dx

3

p

(x2+ 1)2 (DS 3)

46. e

Z

1

4

√

1 + lnx

x dx. (DS 0,8(2

4

√

2−1)) 47.

3

Z

−3

x2

√

9−x2dx. (DS. 81π

8 )

48.

3

Z

0

r

x

6−xdx. (DS

3(π−2)

2 )

Chı’ dˆa˜n. D˘a.t x= sin2t. 49.

4

Z

3

x2+ 3

x−2dx. (DS 11

2 + 7ln2)

50. −1

Z

−2

x+

x2(x−1)dx. (DS ln

4 −

1 2)

51.

1

Z

0

(x2+ 3x)dx

(x+ 1)(x2+ 1) (DS

52.

1

Z

0

dx

√

x2+ 2x+ 2 (DS ln

2 +

√

5 +

√

2)

53.

4

Z

0

dx

1 +√2x+ (DS 2−ln 2) 54.

2

Z

1

e1x

x3dx. (DS

1 2(e−e

1

4))

55. e

Z

1

dx

x(1 + ln2x) (DS π 4)

56. e

Z

1

cos(lnx)

x dx. (DS sin 1)

57.

1

Z

0

xe−xdx. (DS 1−2

e)

58. π/3

Z

π/4

xdx

sin2x (DS

π(9−4

√

3) 36 )

59.

3

Z

1

lnxdx. (DS ln 3−2) 60.

2

Z

1

xlnxdx. (DS ln 2−

4)

61.

1/2

Z

0

arc sinxdx. (DS π 12 +

√

3 −1) 62.

π

Z

0

63. π/2

Z

0

e2xcosxdx. (DS e π −

2 )

64.

2

Z

0

|1−x|dx. (DS 1) 65.

b

Z

a

|x|

x dx. (DS |b| − |a|) T´ınh c´ac t´ıch phˆan sau dˆay

66. a/b

Z

0

dx a2+b2x =

π 4ab

67.

1

Z

0

x2dx

√

4 + 2x =

√

6− 64

15

68.

2

Z

0

dx

x2+ 5x+ 4 =

1 3ln

5

69.

1

Z

0

dx

x2−x+ 1 =

2π 3√3 70.

1

Z

0

(x2+ 1)

x4+x2+ 1dx=

π 2√2 71.

pi/2

Z

0

dx

1 + cosx =

72.

1

Z

0

√

x2+ 1dx= √1

2+

2ln(1 +

√

73.

1

Z

0

1− √

x23/2dx = 3π

32 D˘a.t x= sin3ϕ.

74. a

Z

0

x2

r

a−x a+xdx=

π

4 −

a2, a >0 D˘a.t x=acosϕ.

75.

2a

Z

0

√

2ax−x2dx= πa

2 D˘a.t x= 2asin2ϕ.

76.

1

Z

0

ln(1 +x) +x2 dx=

π 8ln

Chı’ dˆa˜n. D˘a.t x= tgt rˆ` i ´ap du.ng cˆong th´u.co sint+ cost =

√

2 cosπ −t

77. π

Z

0

xsinx

1 + cos2xdx =

π2

4

Chı’ dˆa˜n. Biˆe’u diˆ˜ne π

Z

0

= π/2

Z

0

+ π

Z

π/2

rˆ` i thu c hiˆe.n ph´ep dˆo’i biˆe´n trongo t´ıch phˆan t`u.π/2 dˆe´n π.

78. π

Z

−π

3

√

sinxdx=

79. π

Z

−π

80. π/2

Z

−π/2

(cos2x+x2sinx)dx= π

81.

1

Z

−1

(ex+e−x)tgxdx=

82. pi/2

Z

0

sinxsin 2x sin 3xdx =

83. e

Z

1/e

|lnx|dx = 2(1−e−1) 84.

π

Z

0

excos2xdx= 5(e

π− 1)

85. e

Z

1

(1 + lnx)2dx= 2e−1

Chı’ dˆa˜n. T´ıch phˆan t`u.ng phˆ` n.a

11.3 Mˆo.t sˆo´ ´u.ng du ng cu’ a t´ıch phˆan x´ac di.nh

11.3.1 Diˆe.n t´ıch h`ınh ph˘a’ng v`a thˆe’ t´ıch vˆa.t thˆe’ 1Diˆe.n t´ıch h`ınh ph˘a’ng

x=a, x=bv`a tru.c Ox du.o..c t´ınh theo cˆong th´u.c

SD = b

Z

a

f(x)dx. (11.6)

Nˆe´u f(x)60 ∀x∈[a, b] th`ı SD =−

b

Z

a

f(x)dx (11.6*)

Nˆe´u d´ay h`ınh thang cong n˘a`m trˆen tru.cOy th`ı

SD = d

Z

c

g(y)dy, x=g(y), y ∈[c, d] 2+Nˆe´u du.`o.ng congL

du.o c cho bo.’i phu.o.ng tr`ınh tham sˆo´x=ϕ(t), y=ψ(t),t∈[α, β] th`ı

SD = β

Z

α

ψ(t)ϕ0(t)dt (11.7)

3+ Diˆ

e.n t´ıch cu’a h`ınh qua.t gi´o.i ha.n bo.’i du.`o.ng cong cho du.´o.i da.ng to.a dˆo cu cρ=f(ϕ) v`a c´ac tiaϕ=ϕ0 v`aϕ=ϕ1 du.o c t´ınh theo cˆong

th´u.c

SQ=

ϕ1

Z

ϕ0

[f(ϕ)]2dϕ. (11.8)

4+ Nˆe´u miˆ`ne D ={(x, y) :a6x6b;f1(x)6y 6f2(x)} th`ı

SD = b

Z

a

2 Thˆe’ t´ıch vˆa t thˆe’

1+Nˆe´u biˆe´t du.o..c diˆe.n t´ıchS(x) cu’a thiˆe´t diˆe.n ta.o nˆen bo’ i vˆ a.t thˆe’ v`a m˘a.t ph˘a’ng vuˆong g´oc v´o.i tru.c Ox ta.i diˆe’m c´o ho`anh dˆo. x th`ı xthay dˆo’i mˆo.t da.i lu.o ng b˘a`ng dx th`ı vi phˆan cu’a thˆe’ t´ıch b˘a`ng

dv=S(x)dx,

v`a thˆe’ t´ıch to`an vˆa.t thˆe’ du.o c t´ınh theo cˆong th´u.c

V = b

Z

a

S(x)dx (11.10)

trong d´o [a, b] l`a h`ınh chiˆe´u vuˆong g´oc cu’a vˆa.t thˆe’ lˆen tru.c Ox. 2+ Nˆe´u vˆ

a.t thˆe’ du.o c ta.o nˆen ph´ep quay h`ınh thang cong gi´o.i ha.n bo.’i du.`o.ng cong y = f(x), f(x) > ∀x ∈ [a, b], tru.c Ox v`a c´ac du.`o.ng th˘a’ng x =a, x =b xung quanh tru.c Ox th`ı diˆe.n t´ıch vˆa t thˆe’ tr`on xoay d´o du.o c t´ınh theo cˆong th´u.c

Vx =π b

Z

a

[f(x)]2dx. (11.11)

Nˆe´u quay h`ınh thang cong xung quanh tru.c Oy th`ı vˆa.t tr`on xoay thu du.o c c´o thˆe’ t´ıch

Vy =π d

Z

c

[x(y)]2dy, x=x(y); [c, d] =prOyV. (11.12)

3+ Nˆe´u h`am y=f(x

) du.o c cho bo.’i c´ac phu.o.ng tr`ınh tham sˆo´ x=x(t)

tho’a m˜an nh˜u.ng diˆ`u kiˆe.n n`ao d´o th`ı thˆe’ t´ıch vˆa.t thˆe’ ta.o nˆen bo.’ie ph´ep quay h`ınh thang cong xung quanh tru.c Ox b˘a`ng

Vx =π β

Z

α

y2(t)x0(t)dt (11.13)

4+ Nˆ

e´u h`ınh thang cong du.o c gi´o.i ha.n bo.’i c´ac du.`o.ng cong

y1(x) 6y2(x) ∀x ∈[a, b], d´o y1(x) v`a y2(x) liˆen tu.c trˆen [a, b]

th`ı thˆe’ t´ıch vˆa.t thˆe’ ta.o nˆen ph´ep quay h`ınh thang d´o xung quanh tru.c Oxb˘a`ng

Vx =π b

Z

a

(y2(x))2−(y1(x))2

dx. (11.14)

5+ Dˆo´i v´o.i vˆa.t thˆe’ thu du.o c bo.’i ph´ep quay h`ınh thang cong xung quanh tru.c Oy v`a tho’a m˜an mˆo.t sˆo´ diˆe`u kiˆe.n tu.o.ng tu ta c´o

Vy =π β

Z

α

x2(t)y0(t)dt (11.15)

Vy =π d

Z

c

(x2(y))2−(x1(y))2

dy. (11.16)

C ´AC V´I DU.

V´ı du 1. T`ım diˆe.n t´ıch h`ınh ph˘a’ng gi´o.i ha.n bo.’i du.`o.ng astroid x=acos3t, y=asin3t.

c´ac tru.c to.a dˆo (h˜ay v˜e h`ınh !) nˆen

S = 4S1 =

Z

π/2

asin3t·3acos2t(−sint)dt = 12a2

π/2

Z

0

sin4tcos2tdt

= 2a

2

π/2

Z

0

(1−cos 2t)(1−cos22t)dt = 3πa

3

8 N

V´ı du 2. Trˆen hypecbon x2−y2 =a2 cho diˆe’m M(x0, y0) x0 > 0,

y0 >0 T´ınh diˆe.n t´ıch h`ınh ph˘a’ng gi´o.i ha.n bo.’i tru.c Ox, hypecbˆon v`a

tiaOM

Gia’i. Ta chuyˆe’n sang to.a dˆo cu c theo cˆong th´u.c x = rcosϕ, y=rsinϕ Khi d´o phu.o.ng tr`ınh hypecbˆon c´o da.ng

r2 = a

2

cos2ϕ−sin2

ϕ = a2 cos 2ϕ· D˘a.t tgα= y0

x0

v`a lu.u ´y r˘a`ng x2

0 −y02 =a2 ta thu du.o c

S =

α

Z

0

r2dϕ= a

2

2 α

Z

0

dϕ cos 2ϕ =

a2

4 ln

1 + tgα 1−tgα = a

2

4 ln

(x0+y0)2

a2 =

a2 ln

x0+y0

a · O’ dˆay ta d˜a su.’ du.ng cˆong th´u.c

Z

dt cost = ln

tgt

2 + π

V´ı du 3. T´ınh diˆe.n t´ıch h`ınh ph˘a’ng gi´o.i ha.n bo.’i c´ac du.`o.ng c´o phu.o.ng tr`ınhx2

+y2 = 2y, x2+y2 = 4y; y=x v`ay =−x.

Gia’i. Du.a phu.o.ng tr`ınh du.`o.ng tr`on vˆ` da.ng ch´ınh t˘a´c ta c´o:e x2+ (y−1)2 = v`ax2+ (y−2)2 = D´o l`a hai du.`o.ng tr`on tiˆe´p x´uc

trong ta.i tiˆe´p diˆe’m O(0,0) T`u d´o miˆ`n ph˘a’nge D gi´o.i ha.n bo.’i c´ac du.`o.ng d˜a cho dˆo´i x´u.ng qua tru.c Oy L`o.i gia’i s˜e du.o c do.n gia’n ho.n nˆe´u ta chuyˆe’n sang to.a dˆo cu c (v´o.i tru.c cu c tr`ung v´o.i hu.´o.ng du.o.ng cu’a tru.c ho`anh):

x=rcosϕ y =rsinϕ ⇒

(

x2+y2 = 2y⇒r= sinϕ,

x2+y2 = 4y⇒r= sinϕ,

v`a

D =

n

(r, ϕ) : π

4 6ϕ6 3π

4 ; sinϕ6r64 sinϕ

o

.

K´y hiˆe.uS∗ l`a diˆe.n t´ıch phˆa` n h`ınh tr`on gi´o.i ha.n bo.’i du.`o.ng tr`onx2+ y2 = 4y (t´u.c l`a r = sinϕ) v`a hai tia ϕ = π

4 v`a ϕ = 3π

4 ; S l`a diˆe.n t´ıch phˆan h`ınh tr`on gi´o.i ha.n bo’ i x2+y2 = 2y (t´u.c l`ar = sinϕ) v`a hai tia d˜a nˆeu Khi d´o

SD =S∗−S = 2h1

π/2

Z

π/4

(4 sinϕ)2dϕ−1

2 π/2

Z

π/4

(2 sinϕ)2dϕi

= 12 π/2

Z

π/4

sin2ϕdϕ= 3π

2 + N

V´ı du 4. T´ınh thˆe’ t´ıch vˆa.t tr`on xoay ta.o nˆen ph´ep quay h`ınh thang cong gi´o.i ha.n bo’ i c´ ac du.`o.ng y=±b, x

2

a2 −

y2

b2 = xung quanh

tru.c Oy.

l`a du’ Ta c´o

V = 2V1 = 2π

b

Z

0

x2dy= 2πa2 b

Z

0

1 + y

2

b2

dy

= 2πa2y+ y

3

3b2

b

0

= 3πa

2

b. N

V´ı du 5. T´ınh thˆe’ t´ıch vˆa.t thˆe’ lˆa.p nˆen quay astroid x=acos3t,

y=asin3t, 06t62π xung quanh tru.cOx.

Gia’i. Du.`o.ng astroid dˆo´i x´u.ng dˆo´i v´o.i c´ac tru.c Oxv`a Oy Do d´o Vx =π

a

Z

−a

y2dx= 2π a

Z

0

y2dx

y2 =a2sin6t, dx=−3acos2tsintdt t= π

2 khix= 0, t= khix=a. Do d´o

V = 2π a

Z

0

y2dx=−6a3π

0

Z

π/2

sin6tcos2tsintdt

= 6a3π

0

Z

π/2

(1−cos2t)3cos2t(−sintdt) = 6a3π

0

Z

π/2

(cos2t−3 cos4t+ cos6t−cos8t)(d(cost) =· · ·= 32

105πa

3

. N

V´ı du 6. T´ınh thˆe’ t´ıch vˆa.t thˆe’ gi´o.i ha.n bo.’i hypecboloid mˆo.t tˆa` ng x2

a2 +

y2

b2 −

z2

v`a c´ac m˘a.t ph˘a’ng z = 0, z =h (h >0)

Gia’i. Ta s˜e ´ap du.ng cˆong th´u.c (11.10), d´o ta x´et c´ac thiˆe´t diˆe.n ta.o nˆen bo.’i c´ac m˘a.t ph˘a’ng vuˆong g´oc v´o.i tru.cOz Khi d´o (11.10) c´o da.ng

V = h

Z

0

S(z)dz,

trong d´oS(z) l`a diˆe.n t´ıch cu’a thiˆe´t diˆe.n phu thuˆo.c v`ao z Khi c˘a´t vˆa.t thˆe’ bo.’ i m˘a.t ph˘a’ng z= const ta thu du.o c elip v´o.i phu.o.ng tr`ınh

x2

a2 +

y2

b2 = +

z2

c2

z = const

 

⇔

      

x2

a21 + z

c2

+ y

2

b21 + z

c2

=

z= const T`u d´o suy r˘a`ng

a1 =

r

a21 +z

c2

, b1 =

r

b21 +z

c2

l`a c´ac b´an tru.c cu’a elip Nhu.ng ta biˆe´t r˘a`ng diˆe.n t´ıch h`ınh elip v´o.i b´an tru.ca1,b1 l`aπa1b1 (c´o thˆe’ t´ınh b˘a`ng cˆong th´u.c (11.7) dˆo´i v´o.i elip

c´o phu.o.ng tr`ınh tham sˆo´x=a1cost, y=b1sint, t∈[0,2π])

Nhu vˆa.y

S(z) =πab1 +z

2

c2

, z ∈[0, h] T`u d´o theo cˆong th´u.c (11.10) ta c´o

V = h

Z

0

πab1 + z

2

c2

dz =πabh1 + h

2

3c2

. N

Gia’i. Vˆa.t tr`on xoay thu du.o c c´o t´ınh chˆa´t l`a mo.i thiˆe´t diˆe.n ta.o bo.’ i m˘a.t ph˘a’ng vuˆong g´oc v´o.i tru.c quay dˆe`u l`a v`anh tr`on gi´o.i ha.n bo.’i c´ac du.`o.ng tr`on dˆ` ng tˆam X´et thiˆe´t diˆe.n c´ach gˆo´c to.a dˆo khoa’ng b˘a`ngo y (0 6y64) Ta c´o

S =πR2−πr2 =π[(3 +x)2−(3−x)2] = 12πx= 12πp4−y v`ıx l`a ho`anh dˆo cu’a diˆe’m trˆen parabˆon d˜a cho Khi y thay dˆo’i da.i lu.o ng dy th`ı vi phˆan thˆe’ t´ıch

dv =S(y)dy = 12πp4−ydy. Do d´o thˆe’ t´ıch to`an vˆa.t b˘a`ng

V = 12π

4

Z

0

p

4−ydy= 8π(4−y)3/2

0

= 64π N

V´ı du 8. T`ım thˆe’ t´ıch vˆa.t thˆe’ gi´o.i ha.n bo.’i c´ac m˘a.t x2+y2 = R2; y= 0, z = 0, x

R + z

h−1 = 0, x R −

z

h −1 =

Gia’i. Do t´ınh dˆo´i x´u.ng (h˜ay v˜e h`ınh) cu’a vˆa.t thˆe’ dˆo´i v´o.i m˘a.t ph˘a’ng x = nˆen ta chı’ cˆ` n t´ınh thˆe’ t´ıch phˆaa ` n n˘a`m g´oc phˆa` n t´am th´u nhˆa´t Mo.i thiˆe´t diˆe.n ta.o nˆen bo.’i c´ac m˘a.t ph˘a’ng ⊥Ox dˆ`ue l`a h`ınh ch˜u nhˆa.t ABCD v´o.iOA=x Khi d´o

S(x) =SABCD =AB·AD= h

R(R−x)·

√

R2−x2.

T`u d´o thu du.o..c

V = R

Z

0

S(x)dx= 2h R

R

Z

0

(R−x)

√

R2−x2dx (d˘a.tx=Rsint)

= 2hR2 π/2

Z

0

(1−sint) cos2tdt= hR

2(3π−4)

B `AI T ˆA P

Trong c´ac b`ai to´an sau dˆay (1-17) t´ınh diˆe.n t´ıch c´ac h`ınh ph˘a’ng gi´o.i ha.n bo’ i c´ ac du.`o.ng d˜a chı’

1. y= 6x−x2 −7, y=x−3. (DS.

2) 2. y= 6x−x2, y = (DS 36)

3. 4y= 8x−x2, 4y=x+ 6. (DS 5

24) 4. y= 4−x2,y=x2−2x. (DS 9)

5. 6x=y3−16y, 24x=y3−16y (DS 16) 6. y= 1−ex,x= 2, y = 0. (DS. e2−3)

7. y=x2−6x+ 10,y= 6x−x2; x=−1 (DS 211

3) 8. y= arc sinx,y=±π

2, x= (DS 2) 9. y=ex,y =e−x, x= 1. (DS. (e−1)

2

e ) 10. y2 = 2px, x2 = 2py (DS

3p

2

)

11. x2+y2+ 6x−2y+ = 0, y=x2+ 6x+ 10

(DS S1 =

3π+

6 , S2 =

9π−2 )

12. x=a(t−sint),y=a(1−cost),t ∈[0,2π] (DS 3πa2)

Chı’ dˆa˜n. Dˆay l`a phu.o.ng tr`ınh tham sˆo´ cu’a du.`o.ng xycloid 13. x=acos3t, y=asin3

t,t ∈[0,2π] (DS 3πa

2

8 ) 14. x=acost,y =bsint,t∈[0,2π] (DS πab)

15. Du.`o.ng lemniscate Bernoulliρ2 =a2cos 2ϕ (DS. a2)

16. Du.`o.ng h`ınh tim (Cacdioid)ρ=a(1 + cosϕ). (DS 3πa

2

17∗ C´ac du.`o.ng tr`on ρ=

√

3acosϕ, ρ= 2asinϕ. (DS a25

6π−

√

3)

Trong c´ac b`ai to´an sau (18-22) h˜ay t´ınh thˆe’ t´ıch vˆa.t thˆe’ theo diˆe.n t´ıch c´ac thiˆe´t diˆe.n song song.

18. Thˆe’ t´ıch h`ınh elipxoid x

2

a2 +

y3 b2 +

z2

c2 = (DS

4 3πabc)

19. Thˆe’ t´ıch vˆa.t thˆe’ gi´o.i ha.n bo.’i m˘a.t tru x2+y2 =a2,y2+z2 =a2 (DS 16

3 a

3)

Chı’ dˆa˜n. Do t´ınh dˆo´i x´u.ng, chı’ cˆ` n t´ınh thˆe’ t´ıch mˆo.t phˆaa ` n t´am vˆa.t thˆe’ v´o.ix > 0,y > 0, z >0 l`a du’ C´o thˆe’ lˆa´y c´ac thiˆe´t diˆe.n song song v´o.i m˘a.t ph˘a’ng xOz D´o l`a c´ac h`ınh vuˆong

20. Thˆe’ t´ıch vˆa.t thˆe’ h`ınh n´on v´o.i b´an k´ınh d´ay R v`a chiˆ`u caoe h. (DS πR

2h

3 )

Chı’ dˆa˜n. Di.ch chuyˆe’n h`ınh n´on vˆe` vi tr´ı v´o.i dı’nh ta.i gˆo´c to.a dˆo.

v`a tru.c dˆo´i x´u.ng l`aOx Thiˆe´t diˆe.n cˆa` n t`ım l`a h`ınh tr`on v´o.i b´an k´ınh r(x) = R

xx (?)

21. Thˆe’ t´ıch vˆa.t thˆe’ gi´o.i ha.n bo.’i c´ac m˘a.t n´on (z−2)2 = x

2

3 + y2

2 v`a m˘a.t ph˘a’ngz = (DS 8π

√

6 )

22. Thˆe’ t´ıch vˆa.t thˆe’ gi´o.i ha.n bo.’i m˘a.t tru partabolic z = 4−y2, c´ac

m˘a.t ph˘a’ng to.a dˆo v`a m˘a.t ph˘a’ng x=a. (DS 16a )

Trong c´ac b`ai to´an sau dˆay (23-34) h˜ay t´ınh thˆe’ t´ıch cu’a vˆa.t tr`on xoay thu du.o c bo.’i ph´ep quay h`ınh ph˘a’ng D gi´o.i ha.n bo’ i du.` o.ng (c´ac du.`o.ng) cho tru.´o.c xung quanh tru.c cho tru.´o.c

23. D :y2= 2px,x=a

24. D : x

2

a2 +

y2

b2 61 (b < a) xung quanh tru.c Oy (DS.

4π a

2

b) 25. D : x

2

a2 +

y2

b2 61 (b < a) xung quanh tru.c Ox (DS.

4π ab

2

) 26. D : 2y=x2; 2x+ 2y−3 = xung quanh tru.cOx (DS 18

15π) 27. D :x2+y2 = 1; x+y= xung quanh tru.c Ox (DS. π

3) 28. D :x2+y2 = 4, x=−1,x= 1, y >

0 xung quanh tru.cOx. (DS 8π)

29. D :y= sinx, 0 6x6π, y= xung quanh tru.c Ox (DS. π

2

2 ) 30. D : x

2

a2 −

y2

b2 = 1, y = 0,y=bxung quanh tru.c Oy (DS.

4 3πa

2

b) 31. D :y2+x−4 = 0, x= xung quanh tru.cOy (DS 34

15π) 32. D :xy= 4,y = 0, x= 1, x= xung quanh tru.c Ox (DS 12π) 33. D :x2+ (y−b)2 6R2 (0 < R6b) xung quanh tru.c Ox.

(DS 2π2bR2)

Chı’ dˆa˜n. H`ınh tr`on D c´o thˆe’ xem nhu hiˆe.u cu’a hai thang cong D1 =

(x, y) :−R6 x6R,06y 6− √

R2−x2 v`a

D2 =

(x, y) :−R6 x6R,06y 6+

√

R2−x2 .

34∗. D = (x, y) : 0 6 y 6 √R2−x2 xung quanh du.`o.ng th˘a’ng

y=R.

(DS 3π−4 πR

3

)

Chı’ dˆa˜n. Chuyˆe’n gˆo´c to.a dˆo vˆe` diˆe’m (0, R)

11.3.2 T´ınh dˆo d`ai cung v`a diˆe.n t´ıch m˘a.t tr`on xoay

y=ψ(t) th`ı vi phˆan dˆo d`ai cung du.o c biˆe’u diˆe˜n bo.’i cˆong th´u.c d=p1 + (y0

x)2dx=

q

1 + (x0

y)2dy =

q

x0 t

2

+y0 t

2

dt (11.17) v`a dˆo d`ai cu’a du.`o.ng cong L(A, B) du.o..c t´ınh bo.’i cˆong th´u.c

`(A, B) = xZB=b

xA=a

p

1 + (y0)2dx=

yB

Z

yA

q

1 + (x0 y)2dy

= tB

Z

tA

q

x0 t

2

+y0 t

2

dt. (11.18)

Nˆe´u du.`o.ng cong du.o..c cho bo.’i phu.o.ng tr`ınh to.a dˆo cu cρ=ρ(ϕ) th`ı

d` =

q

ρ2+ ρ0 ϕ

2

dϕ v`a

`(A, B) = ϕB

Z

ϕA

q

ρ2 + ρ0 ϕ

2

dϕ. (11.19)

2+ Nˆe´u m˘a.t σ thu du.o c quay du.`o.ng cong cho trˆen [a, b] bo.’ i h`am khˆong ˆam y = f(x) > 0 xung quanh tru.c Ox th`ı vi phˆan diˆe.n t´ıch m˘a.t

ds= 2π·y+ (y+dy)

2 d`=π(2y+dy)d`≈ 2πyd` v`a diˆe.n t´ıch m˘a.t tr`on xoay du.o c t´ınh theo cˆong th´u.c

Sx= 2π b

Z

a

f(x)p1 + (f0

Nˆe´u quay du.`o.ng cong L(A, B) xung quanh tru.cOy th`ıds≈2πx(y)d` v`a

Sy = 2π yB

Z

yA

x(y)

q

1 + (x0

y)2dy. (11.21) Nˆe´u du.`o.ng congL(A, B) du.o..c cho bo.’i phu.o.ng tr`ınh tham sˆo´x=ϕ(t), y=ψ(t)>0 (t∈[α, β]) th`ı

Sx = 2π β

Z

α ψ(t)

q

ϕ02

+ψ02dt. (11.22)

Tu.o.ng tu ta c´o

Sy = 2π β

Z

α ϕ(t)

q

ϕ02

+ψ02dt, ϕ(t)>0. (11.23)

C ´AC V´I DU. V´ı du 1. T´ınh dˆo d`ai du.`o.ng tr`on b´an k´ınh R.

Gia’i. Ta c´o thˆe’ xem du.`o.ng tr`on d˜a cho c´o tˆam ta.i gˆo´c to.a dˆo Phu.o.ng tr`ınh du.`o.ng tr`on du.´o.i da.ng tham sˆo´ c´o da.ng x = Rcost, y=Rsint,t∈[0,2π] Ta chı’ cˆ` n t´ınh dˆo d`ai cu’a mˆo.t phˆaa ` n tu du.`o.ng tr`on ´u.ng v´o.i 6t6 π

2 l`a du’ Theo cˆong th´u.c (11.18) ta c´o

`= π/2

Z

0

p

(−Rsint)2+ (Rcost)2dt= 4Rt

π/2

0 = 2πR N

V´ı du 2. T´ınh dˆo d`ai cu’a v`ong th´u nhˆa´t cu’a du.`o.ng xo˘a´n ˆo´c Archimedesρ=aϕ.

Gia’i. Theo di.nh ngh˜ıa, du.`o.ng xo˘a´n ˆo´c Archimedes l`a du.`o.ng cong

t`u gˆo´c-cu..c m`a tia n`ay la.i quay xung quanh gˆo´c cu c v´o.i vˆa.n tˆo´c g´oc cˆo´ di.nh V`ong th´u nhˆa´t cu’a du.`o.ng xo˘a´n ˆo´c Archimedes du.o c ta.o nˆen g´oc cu c ϕ biˆe´n thiˆen t`u dˆe´n 2π Do d´o theo cˆong th´u.c (11.19) ta c´o

` =

2π

Z

0

p

a2ϕ2+a2dϕ=a 2π

Z

0

p

ϕ2+ 1dϕ.

T´ıch phˆan t`u.ng phˆ` n b˘a`ng c´ach d˘a.ta u=pϕ2+ 1, dv=dϕta c´o

`=ahϕpϕ2+ 12π

−

2π

Z

0

ϕ2

p

ϕ2+ 1dϕ

i

=a

h

ϕpϕ2+ 1

2π

0

−

2π

Z

0

ϕ2+ 1−1

p

ϕ2+ 1 dϕ

i

=ah1 2ϕ

p

ϕ2 + +1

2ln(ϕ+

p

ϕ2+ 1)i

2π

0

=ahπ

√

4π2+ +

2 2π+

√

4π2+ 1i.N

V´ı du 3. T´ınh diˆe.n t´ıch m˘a.t cˆa` u b´an k´ınh R.

Gia’i. C´o thˆe’ xem m˘a.t cˆa` u c´o tˆam ta.i gˆo´c to.a dˆo v`a thu du.o c bo.’i ph´ep quay nu.’ a du.`o.ng tr`on y=

√

R2−x2 xung quanh tru.c Ox.

Phu.o.ng tr`ınh du.`o.ng tr`on c´o da.ng x2 + y2 = R2 Do d´o y0 =

−√ x

R2−x2 Theo cˆong th´u.c (11.20) ta c´o

Sx = 2π R

Z

−R

√

R2−x2·

s

1 + x

2

R2 −x2dx= 2π

R

Z

−R

√

R2 −x2+x2dx

= 2πRx

R

−R = 4πR

2

. N

Gia’i. Biˆe´n ρ chı’ nhˆa.n gi´a tri thu..c cos 2ϕ > t´u.c l`a

−π/46ϕ6π/4 (nh´anh bˆen pha’i) hay 3π/46ϕ65π/4 (nh´anh bˆen tr´ai) Vi phˆan cung cu’a lemniscat b˘a`ng

d`=

q

ρ2+ρ02

dϕ=

s

a2cos 2ϕ+ (−√asin 2ϕ

cos 2ϕ

2

dϕ = √adϕ

cos 2ϕ·

Ngo`ai ray =ρsinϕ=a√cos 2ϕ·sinϕ T`u d´o diˆe.n t´ıch cˆa` n t`ım b˘a`ng hai lˆ` n diˆe.n t´ıch cu’a m˘a.t thu du.o c bo.’i ph´ep quay nh´anh pha’i Do d´oa theo (11.20)

S = 2·2π π/4

Z

0

yds= 4π π/4

Z

0

a√cos 2ϕ·sinϕ·adϕ

√

cos 2ϕ

= 4π π/4

Z

0

a2sinϕdϕ= 2πa2(2− √

2) N

V´ı du 5. T`ım diˆe.n t´ıch m˘a.t ta.o nˆen bo.’i ph´ep quay cung parabˆon y= x

2

2 , 06x6

√

3 xung quanh tru.cOy.

Gia’i. Ta c´o x = √2y, x0 = √1

2y Do d´o, ´ap du.ng cˆong th´u.c (11.18) ta thu du.o c

S = 2π

3/2

Z

0

p

2y

r

1 +

2ydy = 2π

3/2

Z

0

p

2y+ 1dy

= 2π· (2y+ 1)

3/2

3

3/2

= 14π · N

V´ı du 6. T`ım diˆe.n t´ıch m˘a.t ta.o nˆen bo.’i ph´ep quay elipx2+ 4y2 = 26

xung quanh: a) tru.cOx; b) tru.c Oy.

Gia’i. Nu.’ a trˆen cu’a elip d˜a cho c´o thˆe’ xem nhu dˆ` thi cu’a h`amo y=

2

√

c`on trˆen khoa’ng (−6,6) da.o h`am khˆong bi ch˘a.n Do vˆa.y khˆong thˆe’ t´ınh b˘a`ng cˆong th´u.c (11.20) to.a dˆo Dˆe` c´ac du.o c.

Dˆe’ kh˘a´c phu.c kh´o kh˘an d´o, ta d`ung ph´ep tham sˆo´ h´oa du.`o.ng elip: x = cost, y= sint, 06t62π

1+Ph´ep quay xung quanh tru.cOx Ta x´et nu.’ a trˆen cu’a elip tu.o.ng ´

u.ng v´o.i 06t6π Theo cˆong th´u.c (11.22) du.´o.i da.ng tham sˆo´ ta c´o

Sx = 2π π

Z

0

3 sint·p36 sin2t+ cos2tdt.

D˘a.t cost = √2

3sinϕ ta c´o

Sx= 24

√

3π π/3

Z

−π/3

cos2ϕdϕ=

√

3π(4π+

√

3)

2+ Ph´

ep quay xung quanh tru.c Oy Ta x´et nu.’ a bˆen pha’i cu’a elip (tu.o.ng ´u.ng v´o.it ∈−π

2, π

i

Tu.o.ng tu nhu trˆen ta ´ap du.ng (11.23) v`a thu du.o c

Sy = 2π π/2

Z

−π/2

6 cost·p36 sin2t+ cos2tdt D˘a.t sint = √1

3shϕ

= 24

√

3π

arcshZ √3

−arcsh√3

ch2ϕdϕ= 24

√

3π

√

3 + ln(2 +

√

3). N

B `AI T ˆA P T´ınh dˆo d`ai cung cu’a du.`o.ng cong 1. y=x3/2 t`u.x= dˆe´n x= (DS

27(10

√

2. y=x2−1 t`u.x=−1 dˆe´n x= (DS. √5 +

2ln(2 +

√

5)) 3. y= a

2 e

x/a+e−x/a t`u.x= dˆe´n x=a (DS. a(e

2−1)

2e ) 4. y= ln cosx t`u.x= dˆe´n x= π

6 (DS 2ln 3) 5. y= ln sinx t`u.x = π

3 dˆe´nx = 2π

3 (DS ln 3) 6. x=etsint,y =etcost, 06t 6 π

2 (DS

√

2(eπ/2−1))

7. x=a(t−sint),y=a(1−cost); 06t62π (DS 8a) 8. x=acos3t,y=asin3

t; 06t62π (DS 6a)

Chı’ dˆa˜n. V`ıpx0 t

2

+y0 t

2

= 3a

1 |sin 2t|v`a h`am|sin 2t|c´o chu k`yπ/2 nˆen `=

π/2

Z

0

d`.

9. x=etcost, y=etsint t`u.t = dˆe´nt = lnπ (DS.

√

2(π−1)) 10. x = sint+ cost, y= sint−8 cost t`u.t = dˆe´n t= π

2 (DS 5π)

11. ρ=aekθ (du.`o.ng xo˘´n ˆo´c lˆoga) t`a u.θ = dˆe´nθ =T. (DS a

k

√

1 +k2(ekT −1))

12. ρ=a(1−cosϕ),a >0, 06ϕ62π (du.`o.ng h`ınh tim) (DS 8a) 13∗. ρϕ = t`u diˆe’m A2,1

2

dˆe´n diˆe’m B1 2,2

- du.`o.ng xo˘a´n ˆo´c hypecbon

(DS

√

5 + ln

3 +

√

5 )

T´ınh diˆe.n t´ıch c´ac m˘a.t tr`on xoay thu du.o c quay cung du.`o.ng cong hay du.`o.ng cong xung quanh tru.c cho tru.´o.c.

14. Cung cu’a du.`o.ng y = x3 t`u.x= −2

3 dˆe´n x =

(DS 2π 27

125

27 −1

) 15. Du.`o.ng x=acos3t, y=asin3

t xung quanh tru.cOx. (DS 12

5 πa

2

) 16. x

2

a2 +

y2

b2 = 1, a > b xung quanh tru.c Ox.

(DS 2πb

b+a

εarc sinε

, ε l`a tˆam sai cu’a elip)

Chı’ dˆa˜n. Da.o h`am hai vˆe´ phu.o.ng tr`ınh elip rˆo`i r´ut yy0 =

−bx

2

a2 , c`on biˆe’u th´u.c du.´o.i dˆa´u t´ıch phˆan du.o c viˆe´t y

p

1 +y02

dx =

p

y2+ (yy0)3dx.

17. Cung du.`o.ng tr`on x2 + (y−b)2 = R (khˆong c˘a´t tru.c Oy) t`u.y

dˆe´n y2 xung quanh tru.cOy (DS 2πR(y2−y1))

Chı’ dˆa˜n. M˘a.t thu du.o c l`a d´o.i cˆ` ua

18. y= sinxt`u.x= dˆe´n x=π xung quanh tru.cOx. (DS 2π

√

2 + ln(1 +

√

2)) 19. y= x

3

3 t`u.x=−2 dˆe´n x= xung quanh tru.c Ox. (DS 34

√

17−2

9 π)

20. Cung bˆen tr´ai du.`o.ng th˘a’ng x = cu’a du.`o.ng cong y2 = +x, xung quanh tru.cOx (DS. 62π

3 ) 21. y= a

2 e

x/a+e−x/a t`u.x= dˆe´nx=a (a >0). (DS πa

2

4 (e

2+ 4−e−2))

22. y2 = 4x t`u.x= dˆe´n x

= 3, xung quanh tru.c Ox (DS. 56π ) 23. x=etsint, y=etcost t`u.t= dˆe´nt = π

2, xung quanh tru.cOx. (DS 2π

√

2 (e

24. x=acos3t, y=asin3t, 06t62π; quay xung quanh tru.cOx. (DS 12

5 πa

2

)

Chı’ dˆa˜n. V`ı du.`o.ng cong c´o t´ınh dˆo´i x´u.ng qua c´ac tru.c to.a dˆo nˆen chı’ cˆ` n t´ınh diˆe.n t´ıch ta.o nˆen bo.’i mˆo.t phˆaa ` n tu du.`o.ng thuˆo.c g´oc I quay xung quanh tru.c Ox.

25. x = t −sint, y = 1−cost (diˆe.n t´ıch du.o c ta.o th`anh quay mˆo.t cung); xung quanh tru.c Ox.

(DS 64π ) 26. y= sin 2x t`u.x= dˆe´n x= π

2; xung quanh tru.cOx. (DS π

2

2

√

5 + ln(

√

5 + 2)) 27. 3x2+ 4y2

= 12; xung quanh tru.c Oy (DS 2π(4 + ln 3)) 28. x2 =y+ 4, y

= 2; xung quanh tru.c Oy (DS. 62π )

29. Cung cu’a du.`o.ng tr`on x2+y2 = (y >0) gi˜u.a hai diˆe’m c´o ho`anh

dˆo. x=−1 v`a x= 1; xung quanh tru.c Ox (DS 8π)

30. Du.`o.ng h`ınh tim (cacdiod) ρ = a(1 + cosϕ); quay xung quanh tru.c cu c

(DS 32πa

2

5 )

31. Du.`o.ng tr`onρ= 2rsinϕ; quay xung quanh tru.c cu..c (DS 4π2r2)

32. Cung _

AB cu’a du.`o.ng xicloid x = a(t−sint), y = a(1−cost); quay xung quanh du.`o.ng th˘a’ng y=a (DS 16√2πa

2

3 )

Chı’ dˆa˜n. Ap du.ng cˆong th´u´ c

S = 2π

3π

Z

π/2

2(y(t)−a)

q

x0 t

2

+y0 t

2

11.4 T´ıch phˆan suy rˆo.ng

11.4.1 T´ıch phˆan suy rˆo ng cˆa n vˆo ha.n

1 Gia’ su.’ h`am f(x) x´ac di.nh ∀x>a v`a kha’ t´ıch trˆen mo.i doa.n [a, b]. Nˆe´u tˆ` n ta.i gi´o.i ha.n h˜u.u ha.no

lim b→+∞

b

Z

a

f(x)dx (11.24) th`ı gi´o.i ha.n d´o du.o c go.i l`a t´ıch phˆan suy rˆo.ng cu’a h`am f(x) trˆen khoa’ng [a,+∞) v`a k´y hiˆe.u l`a

+∞

Z

a

f(x)dx.

Trong tru.`o.ng ho p n`ay ngu.`o.i ta c`on n´oi r˘a`ng t´ıch phˆan suy rˆo.ng (11.24) hˆo.i tu v`a h`amf(x) kha’ t´ıch theo ngh˜ıa suy rˆo.ng trˆen khoa’ng [a,+∞) Nˆe´u gi´o.i ha.n (11.24) khˆong tˆo` n ta.i th`ı t´ıch phˆan

+∞

Z

a

f(x)dx du.o c go.i l`a t´ıch phˆan phˆan k`y v`a h`amf(x) khˆong kha’ t´ıch theo ngh˜ıa suy rˆo.ng trˆen [a,+∞)

Tu.o.ng tu nhu trˆen, theo di.nh ngh˜ıa b

Z

−∞

f(x)dx = lim a→−∞

b

Z

a

f(x)dx (11.25)

+∞

Z

−∞

f(x)dx= c

Z

−∞

f(x)dx+

+∞

Z

c

f(x)dx, c∈R (11.26) C´ac cˆong th´u.c co ba’n dˆo´i v´o.i t´ıch phˆan suy rˆo.ng

1) T´ınh tuyˆe´n t´ınh Nˆe´u c´ac t´ıch phˆan suy rˆo.ng

+∞

Z

a

f(x)dx v`a

+∞

Z

a

g(x)dxhˆo.i tu. ∀α, β ∈Rth`ı t´ıch phˆan

+∞

Z

a

v`a

+∞

Z

a

(αf(x) +βg(x))dx=α

+∞

Z

a

f(x)dx+β

+∞

Z

a

g(x)dx.

2)Cˆong th´u.c Newton-Leibnitz Nˆe´u trˆen khoa’ng [a,+∞) h`amf(x) liˆen tu.c v`a F(x),x∈[a,+∞) l`a nguyˆen h`am n`ao d´o cu’a n´o th`ı

+∞

Z

a

f(x)dx=F(x)a+∞=F(+∞)−F(a) d´o F(+∞) = lim

x→+∞F(x)

3) Cˆong th´u.c dˆo’i biˆe´n Gia’ su.’ f(x), x∈[a,+∞) l`a h`am liˆen tu.c, ϕ(t), t ∈[α, β] l`a kha’ vi liˆen tu.c v`a a =ϕ(α) ϕ(t) < lim

t→β−0ϕ(t) =

+∞ Khi d´o:

+∞

Z

a

f(x)dx= β

Z

α

f(ϕ(t))ϕ0(t)dt (11.27) 4)Cˆong th´u.c t´ıch phˆan t`u.ng phˆ` n.a Nˆe´uu(x) v`av(x),x∈[a,+∞) l`a nh˜u.ng h`am kha’ vi liˆen tu.c v`a lim

x→+∞(uv) tˆ` n ta.i th`ı:o

+∞

Z

a

udv=uv+a∞−

+∞

Z

a

vdu (11.28)

trong d´o uv+a∞= lim x→+∞(uv)

−u(a)v(a). C´ac diˆ`u kiˆe e.n hˆo.i tu.

1) Tiˆeu chuˆa’n Cauchy T´ıch phˆan

+∞

Z

a

f(x)dx hˆo.i tu v`a chı’ khi

∀ε >0, ∃b=b(ε)>a cho∀b1 > b v`a∀b2 > b ta c´o:

b2

Z

b1

f(x)dx

2) Dˆa´u hiˆe.u so s´anh I Gia’ su.’ g(x) > f(x) > ∀x > a v`a f(x), g(x) kha’ t´ıch trˆen mo.i doa.n [a, b],b <+∞ Khi d´o:

(i) Nˆe´u t´ıch phˆan

+∞

Z

a

g(x)dx hˆo.i tu th`ı t´ıch phˆan

+∞

Z

a

f(x)dx hˆo.i tu

(ii) Nˆe´u t´ıch phˆan

+∞

Z

a

f(x)dxphˆan k`y th`ı t´ıch phˆan

+∞

Z

a

g(x)dxphˆan k`y

3) Dˆa´u hiˆe.u so s´anh II Gia’ su.’ f(x)>0,g(x)>0 ∀x>a v`a lim

x→+∞ f(x) g(x) =λ. Khi d´o:

(i) Nˆe´u < λ < +∞ th`ı c´ac t´ıch phˆan

+∞

Z

a

f(x)dx v`a

+∞

Z

a

g(x)dx dˆ` ng th`o.i hˆo.i tu ho˘a.c dˆoo ` ng th`o.i phˆan k`y

(ii) Nˆe´u λ = v`a t´ıch phˆan

+∞

Z

a

g(x)dx hˆo.i tu th`ı t´ıch phˆan

+∞

Z

a

f(x)dx hˆo.i tu

(iii) Nˆe´u λ = +∞ v`a t´ıch phˆan

+∞

Z

a

f(x)dx hˆo.i tu th`ı t´ıch phˆan

+∞

Z

a

g(x)dx hˆo.i tu

Dˆe’ so s´anh ta thu.`o.ng su.’ du.ng t´ıch phˆan

+∞

Z

a dx xα

% &

hˆo.i tu nˆe´u α >1, phˆan k`y nˆe´u α61

D- i.nh ngh˜ıa. T´ıch phˆan

+∞

Z

a

f(x)dx du.o c go.i l`a hˆo.i tu tuyˆe.t dˆo´i nˆe´u

t´ıch phˆan

+∞

Z

a

|f(x)|dx hˆo.i tu v`a du.o c go.i l`a hˆo.i tu c´o diˆe`u kiˆe.n nˆe´u t´ıch phˆan

+∞

Z

a

f(x)dx hˆo.i tu nhu.ng t´ıch phˆan

+∞

Z

a

|f(x)|dx phˆan k`y Mo.i t´ıch phˆan hˆo.i tu tuyˆe.t dˆo´i dˆe`u hˆo.i tu

3) T`u dˆa´u hiˆe.u so s´anh II v`a (11.29) r´ut ra

Dˆa´u hiˆe.u thu..c h`anh. Nˆe´u x → +∞ h`am du.o.ng f(x) l`a vˆo c`ung b´e cˆa´p α >0 so v´o.i

x th`ı (i) t´ıch phˆan

+∞

Z

a

f(x)dx hˆo.i tu khi α >1;

(ii) t´ıch phˆan

+∞

Z

a

f(x)dx phˆan k`y α 61

C ´AC V´I DU. V´ı du 1. T´ınh t´ıch phˆan

I =

+∞

Z

2

dx x2√x2−1·

Gia’i. Theo di.nh ngh˜ıa ta c´o

+∞

Z

2

dx

x2√x2−1 = limb→+∞ b

Z

2

dx x2√x2−1·

D˘a.t x =

I(b) = b

Z

2

dx

x2√x2−1 = 1/b

Z

1/2

−dt t2·

t2

r

1 t2 −1

=−

1/b

Z

1/2

tdt

√

1−t2

=

√

1−t2 1/b

1/2

=

r

1−

b2 −

r

1−

4. T`u d´o suy r˘a`ng I = lim

b→+∞I(b) = 2−

√

3

2 Nhu vˆa.y t´ıch phˆan d˜a cho hˆo.i tu N

V´ı du 2. Kha’o s´at su hˆo.i tu cu’a t´ıch phˆan

+∞

Z

1

2x2+ 1

x3+ 3x+ 4dx.

Gia’i. H`am du.´o.i dˆa´u t´ıch phˆan >0∀x>1 Ta c´o f(x) = 2x

2

+ x3+ 3x+ 4 =

2 + x2

x+ x+

4 x2

·

V´o.i x du’ l´o.n h`am f(x) c´o d´ang diˆe.u nhu.

x Do d´o ta lˆa´y h`am ϕ(x) =

x dˆe’ so s´anh v`a c´o lim

x→+∞ f(x)

ϕ(x) = limx→+∞

(2x2+ 1)x

x2+ 3x+ 4 = 26=

V`ı t´ıch phˆan ∞

Z

1

dx

x phˆan k`y nˆen theo dˆa´u hiˆe.u so s´anh II t´ıch phˆan d˜a cho phˆan k`y N

V´ı du 3. Kha’o s´at su. hˆo.i tu cu’a t´ıch phˆan ∞

Z

2

dx

3

√

Gia’i. Ta c´o bˆa´t d˘a’ng th´u.c

3

√

x3−1 >

1

x khix >2

Nhu.ng t´ıch phˆan ∞

Z

2

dx

x phˆan k`y, d´o theo dˆa´u hiˆe.u so s´anh I t´ıch phˆan d˜a cho phˆan k`y N

V´ı du 4. Kha’o s´at su hˆo.i tu v`a d˘a.c t´ınh hˆo.i tu cu’a t´ıch phˆan

+∞

Z

1

sinx x dx.

Gia’i. Dˆ` u tiˆen ta t´ıch phˆan t`a u.ng phˆ` n mˆo.t c´ach h`ınh th´u.ca

+∞

Z

1

sinx

x dx=− cosx

x

+∞

1

−

+∞

Z

1

cosx

x2 dx= cos 1− +∞

Z

1

cosx x2 dx.

(11.30)

T´ıch phˆan

+∞

Z

1

cosx

x2 dx hˆo.i tu tuyˆe.t dˆo´i, d´o n´o hˆo.i tu Nhu vˆa.y

ca’ hai sˆo´ ha.ng o.’ vˆe´ pha’i (11.30) h˜u.u ha.n T`u d´o suy ph´ep t´ıch phˆan t`u.ng phˆ` n d˜a thu c hiˆe.n l`a ho p l´y v`a vˆe´ tr´ai cu’a (11.30) l`a t´ıcha phˆan hˆo.i tu

Ta x´et su hˆo.i tu tuyˆe.t dˆo´i Ta c´o

|sinx|> sin2x= 1−cos 2x v`a vˆa.y ∀b >1 ta c´o

b

Z

1

|sinx|

x dx>

b

Z

1

dx x −

1

b

Z

1

cos 2x

T´ıch phˆan th´u nhˆa´t o.’ vˆe´ pha’i cu’a (11.31) phˆan k`y T´ıch phˆan th´u hai o.’ vˆe´ pha’i d´o hˆo.i tu (diˆe`u d´o du.o c suy b˘a`ng c´ach t´ıch phˆan t`u.ng phˆ` n nhu (11.30)) Qua gi´o.i ha.n (11.31) khia b → +∞ ta c´o vˆe´ pha’i cu’a (11.31) dˆ` n dˆe´na ∞ v`a d´o t´ıch phˆan vˆe´ tr´ai cu’a (11.31) phˆan k`y, t´u.c l`a t´ıch phˆan d˜a cho hˆo.i tu c´o diˆe`u kiˆe.n (khˆong tuyˆe.t dˆo´i). N

B `AI T ˆA P T´ınh c´ac t´ıch phˆan suy rˆo.ng cˆa.n vˆo ha.n 1.

∞

Z

0

xe−x2dx (DS 2)

2. ∞

Z

0

dx x

√

x2−1 (DS

π 6)

3. ∞

Z

0

dx

(x2+ 1)2 (DS

π−2 )

4. ∞

Z

0

xsinxdx. (DS Phˆan k`y)

5. ∞

Z

−∞ 2xdx

x2 + 1 (DS Phˆan k`y)

6. ∞

Z

0

e−xsinxdx. (DS 2)

7.

+∞

Z

2

1

x2−1 +

2 (x+ 1)2

dx. (DS +

1 2ln 3)

8.

+∞

Z

−∞

dx

x2 + 4x+ 9 (DS

π

√

9.

+∞

Z

√

2

xdx

(x2+ 1)3 (DS

1

36) Chı’ dˆa˜n. D˘a.t x=

√

t.

10.

+∞

Z

1

dx x

√

x2+x+ 1 (DS ln

1 +√2

3

) Chı’ dˆa˜n. D˘a.tx= t

11.

+∞

Z

1

arctgx

x2 dx. (DS

π +

ln 2 )

12.

+∞

Z

3

2x+

x2+ 3x−10dx. (DS Phˆan k`y)

13. ∞

Z

0

e−axsinbxdx,a >0 (DS b a2+b2)

14.

+∞

Z

0

e−axcosbxdx, a >0 (DS a a2+b2)

Kha’o s´at su. hˆo.i tu cu’a c´ac t´ıch phˆan suy rˆo.ng cˆa.n vˆo ha.n 15.

∞

Z

1

e−x

x dx. (DS Hˆo.i tu.)

Chı’ dˆa˜n. Ap du.ng bˆa´t d˘a’ng th´u´ c e

−x x 6e

−x ∀ x>1

16.

+∞

Z

2

xdx

√

x4 + 1 (DS Phˆan k`y)

Chı’ dˆa˜n. Ap du.ng bˆa´t d˘a’ng th´u´ c

x

√

x4+ 1 >

x

√

x4+x4 ∀x>

17.

+∞

Z

1

sin23x

3

√

18.

+∞

Z

1

dx

√

4x+ lnx (DS Phˆan k`y)

19.

+∞

Z

1

ln1 +1 x

xα dx. (DS Hˆo.i tu nˆe´u α >0)

20.

+∞

Z

0

xdx

3

√

x5+ 2 (DS Hˆo.i tu.)

21.

+∞

Z

1

cos 5x−cos 7x

x2 dx. (DS Hˆo.i tu.)

22.

+∞

Z

0

xdx

3

√

1 +x7 (DS Hˆo.i tu.)

23.

+∞

Z

0

√

x+

1 + 2√x+x2dx. (DS Hˆo.i tu.)

24. ∞

Z

1

1

√

x(e

1/x−

1)dx (DS Hˆo.i tu.)

25. ∞

Z

1

x+√x+ x2+ 2√5

x4+ 1dx. (DS Phˆan k`y)

26. ∞

Z

3

dx

p

x(x−1)(x−2) (DS Hˆo.i tu.) 27∗.

∞

Z

0

(3x4−x2)e−x2dx. (DS Hˆo.i tu.)

Chı’ dˆa˜n. So s´anh v´o.i t´ıch phˆan hˆo.i tu.

+∞

Z

0

e−x

2

2 dx(ta.i ?) v`a ´ap

28∗.

+∞

Z

5

ln(x−2)

x5 +x2+ 1dx. (DS Hˆo.i tu.)

Chı’ dˆa˜n. Ap du.nng hˆe th´u.c´

lim t→+∞

lnt

tα = ∀α >0⇒x→lim+∞

ln(x−2)

xα = ∀α >0 T`u d´o so s´anh t´ıch phˆan d˜a cho v´o.i t´ıch phˆan hˆo.i tu.

+∞

Z

5

dx

xα, α > Tiˆe´p dˆe´n ´ap du.ng dˆa´u hiˆe.u so s´anh II.

11.4.2 T´ıch phˆan suy rˆo ng cu’ a h`am khˆong bi ch˘a.n Gia’ su.’ h`am f(x) x´ac di.nh trˆen khoa’ng [a, b) v`a kha’ t´ıch trˆen mo.i doa.n [a, ξ],ξ < b Nˆe´u tˆ` n ta.i gi´o.i ha.n h˜u.u ha.no

lim ξ→b−0

ξ

Z

0

f(x)dx (11.32)

th`ı gi´o.i ha.n d´o du.o c go.i l`a t´ıch phˆan suy rˆo.ng cu’a h`amf(x) trˆen [a, b) v`a k´y hiˆe.u l`a:

b

Z

a

f(x)dx. (11.33)

Trong tru.`o.ng ho p n`ay t´ıch phˆan suy rˆo.ng (11.33) du.o c go.i l`a t´ıch phˆan hˆo.i tu Nˆe´u gi´o.i ha.n (11.32) khˆong tˆo` n ta.i th`ı t´ıch phˆan suy rˆo.ng (11.33) phˆan k`y.

Di.nh ngh˜ıa t´ıch phˆan suy rˆo.ng cu’a h`amf(x) x´ac di.nh trˆen khoa’ng (a, b] du.o..c ph´at biˆe’u tu.o.ng tu

doa.n [a, b] v`a tru.`o.ng ho..p n`ay t´ıch phˆan suy rˆo.ng du.o c x´ac di.nh bo.’ i d˘a’ng th´u.c:

b

Z

a

f(x)dx= c

Z

a

f(x)dx+ b

Z

c

f(x)dx.

2 C´ac cˆong th´u.c co ba’n

1) Nˆe´u c´ac t´ıch phˆan b

Z

a

f(x)dx v`a b

Z

a

g(x)dx hˆo.i tu th`ı∀α, β ∈R ta c´o t´ıch phˆan

b

Z

a

[αf(x) +βg(x)]dx hˆo.i tu v`a

b

Z

a

[αf(x) +βg(x)]dx=α b

Z

a

f(x)dx+β b

Z

a

g(x)dx.

2) Cˆong th´u.c Newton-Leibnitz Nˆe´u h`am f(x), x∈ [a, b) liˆen tu.c v`a F(x) l`a mˆo.t nguyˆen h`am n`ao d´o cu’a f trˆen [a, b) th`ı:

b

Z

a

f(x)dx=F(x)ab−0 =F(b−0)−F(a), F(b−0) = lim

x→b−0F(x)

3) Cˆong th´u.c dˆo’i biˆe´n Gia’ su.’ f(x) liˆen tu.c trˆen [a, b) c`on ϕ(t), t ∈ [α, β) kha’ vi liˆen tu.c v`a a = ϕ(α) ϕ(t) < lim

t→β−0ϕ(t) =b Khi

d´o:

b

Z

a

f(x)dx= β

Z

α

4) Cˆong th´u.c t´ıch phˆan t`u.ng phˆ` n Gia’ su.a ’ u(x),x∈[a, b) v`av(x), x∈[a, b) l`a nh˜u.ng h`am kha’ vi liˆen tu.c v`a lim

x→b−0(uv) tˆ` n ta.i Khi d´o;o

b

Z

a

udv =uvba−

b

Z

a vdu

uvba = lim x→b−0(uv)

−u(a)v(a). C´ac diˆ`u kiˆe e.n hˆo.i tu.

1) Tiˆeu chuˆa’n Cauchy Gia’ su.’ h`am f(x) x´ac di.nh trˆen khoa’ng [a, b), kha’ t´ıch theo ngh˜ıa thˆong thu.`o.ng trˆen mo.i doa.n [a, ξ], ξ < b v`a khˆong bi ch˘a.n lˆan cˆa.n bˆen tr´ai cu’a diˆe’m x = b Khi d´o t´ıch phˆan

b

Z

a

f(x)dx hˆo.i tu v`a chı’ khi ∀ε > 0, ∃η ∈[a, b) cho

∀η1, η2 ∈(η, b) th`ı

η2

Z

η1

f(x)dx

< ε.

2) Dˆa´u hiˆe.u so s´anh I. Gia’ su.’ g(x) >f(x) > trˆen khoa’ng [a, b) v`a kha’ t´ıch trˆen mˆo˜i doa.n [a, ξ], ξ < b Khi d´o:

(i) Nˆe´u t´ıch phˆan b

Z

a

g(x)dx hˆo.i tu th`ı t´ıch phˆan b

Z

a

f(x)dxhˆo.i tu

(ii) Nˆe´u t´ıch phˆan b

Z

a

f(x)dxphˆan k`y th`ı t´ıch phˆan b

Z

a

g(x)dx phˆan k`y

3) Dˆa´u hiˆe.u so s´anh II. Gia’ su.’ f(x)>0, g(x)>0,x∈[a, b) v`a lim

x→b−0

(i) Nˆe´u 0< λ <+∞ th`ı c´ac t´ıch phˆan b

Z

a

f(x)dxv`a b

Z

a

g(x)dx dˆ` ngo th`o.i hˆo.i tu ho˘a.c dˆo` ng th`o.i phˆan k`y

(ii) Nˆe´uλ = v`a t´ıch phˆan b

Z

a

g(x)dxhˆo.i tu th`ı t´ıch phˆan b

Z

a

f(x)dx hˆo.i tu

(iii) Nˆe´u λ = +∞ v`a t´ıch phˆan b

Z

a

f(x)dx hˆo.i tu th`ı t´ıch phˆan b

Z

a

g(x)dx hˆo.i tu

Dˆe’ so s´anh ta thu.`o.ng su.’ du.ng t´ıch phˆan: b

Z

a

dx (b−x)α

% &

hˆo.i tu nˆe´u α <1 phˆan k`y nˆe´uα> ho˘a.c

b

Z

a

dx (x−a)α

% &

hˆo.i tu nˆe´u α <1 phˆan k`y nˆe´u α>1

D- i.nh ngh˜ıa. T´ıch phˆan b

Z

a

f(x)dx du.o c go.i l`a hˆo.i tu tuyˆe.t dˆo´i nˆe´u

t´ıch phˆan b

Z

a

|f(x)|dxhˆo.i tu v`a du.o c go.i l`a hˆo.i tu c´o diˆe`u kiˆe.n nˆe´u t´ıch phˆan

b

Z

a

f(x)dxhˆo.i tu nhu.ng b

Z

a

Dˆa´u hiˆe.u thu..c h`anh. Nˆe´u x → b−0 h`am f(x) > x´ac di.nh v`a liˆen tu.c [a, b) l`a vˆo c`ung l´o.n cˆa´p α so v´o.i

b−x th`ı (i) t´ıch phˆan

b

Z

a

f(x)dx hˆo.i tu khiα <1;

(ii) t´ıch phˆan b

Z

a

f(x)dx phˆan k`y α>1

C ´AC V´I DU.

V´ı du 1. X´et t´ıch phˆan

1

Z

0

dx

√

1−x2

Gia’i. H`am f(x) = √

1−x2 liˆen tu.c v`a d´o n´o kha’ t´ıch trˆen mo.i

doa.n [0,1−ε],ε >0, nhu.ng x→1−0 th`ıf(x)→+∞ Ta c´o lim

ε→0 1−ε

Z

0

dx

√

1−x2 = limε→0arc sin(1

−ε) = asrc sin = π 2· Nhu vˆa.y t´ıch phˆan d˜a cho hˆo.i tu N

V´ı du 2. Kha’o s´at su hˆo.i tu cu’a t´ıch phˆan

1

Z

0

√

xdx

√

1−x4 ·

Gia’i. H`am du.´o.i dˆa´u t´ıch phˆan c´o gi´an doa.n vˆo c`ung ta.i diˆe’m x= Ta c´o

√

x

√

1−x4

1

√

1−x ∀x∈[0,1) Nhu.ng t´ıch phˆan

1

Z

0

dx

√

V´ı du 3. Kha’o s´at su. hˆo.i tu cu’a t´ıch phˆan

1

Z

0

dx ex−cosx·

Gia’i. O’ dˆay h`am du.´o.i dˆa´u t´ıch phˆan c´o gi´an doa.n vˆo c`ung ta.i diˆe’m x= Khi x∈(0,1] ta c´o

1

ex−cosx > xe

v`ı r˘a`ng xe>ex−cosx

(ta.i ?) Nhu.ng t´ıch phˆan

1

Z

0

1

xedx phˆan k`y nˆen t´ıch phˆan d˜a cho phˆan k`y N

V´ı du 4. Kha’o s´at su. hˆo.i tu cu’a t´ıch phˆan

+∞

Z

0

arctgx

xα dx, α>

Gia’i. Ta chia khoa’ng lˆa´y t´ıch phˆan l`am hai cho khoa’ng th´u nhˆa´t h`am c´o bˆa´t thu.`o.ng ta.i diˆe’mx= Ch˘a’ng ha.n ta chia th`anh hai nu.’ a khoa’ng (0,1] v`a [1,+∞) Khi d´o ta c´o

+∞

Z

0

arctgx xα dx=

1

Z

0

arctgx xα dx+

+∞

Z

0

arctgx

xα dx. (11.34)

Dˆ` u tiˆen x´et t´ıch phˆana

1

Z

0

arctgx

xα dx, Ta c´o

f(x) = arctgx xα (x∼→0)

x xα =

1

T´ıch phˆan

1

Z

0

ϕ(x)dx hˆo.i tu khi α−1 < ⇒ α < Do d´o t´ıch phˆan

1

Z

0

f(x)dx c˜ung hˆo.i tu khi α <2 theo dˆa´u hiˆe.u so s´anh II.

X´et t´ıch phˆan ∞

Z

1

f(x)dx ´Ap du.ng dˆa´u hiˆe.u so s´anh II 1◦ ta d˘a.t ϕ(x) =

xα v`a c´o lim x→+∞

f(x)

ϕ(x) = limx→+∞

xαarctgx xα =

π · V`ı t´ıch phˆan

∞

Z

0

dx

xα hˆo.i tu α > nˆen v´o.i α > t´ıch phˆan du.o c x´et hˆo.i tu Nhu vˆa.y ca’ hai t´ıch phˆan o.’ vˆe´ pha’i (11.34) chı’ hˆo.i tu 1< α <2

D´o ch´ınh l`a diˆ`u kiˆe.n hˆo.i tu cu’a t´ıch phˆan d˜a cho.e N

V´ı du 5. Kha’o s´at su hˆo.i tu cu’a t´ıch phˆan

1

Z

0

ln(1 +

√

x2)

√

xsin√x dx.

Gia’i. H`am du.´o.i dˆa´u t´ıch phˆan khˆong bi ch˘a.n lˆan cˆa.n pha’i cu’a diˆe’m x= Khi x→0 + ta c´o

ln(1 +

√

x2)

√

xsin√x (x→∼0+0)

3

√

x2

x =

3

√

x =ϕ(x).

V`ı t´ıch phˆan

1

Z

0

dx

3

√

B `AI T ˆA P T´ınh c´ac t´ıch phˆan suy rˆo.ng sau. 1.

6

Z

2

dx

3

p

(4−x)2 (DS

3

√

2)

2.

2

Z

0

dx

3

p

(x−1)2 (DS 6)

3. e

Z

1

dx

xlnx (DS Phˆan k`y)

4.

2

Z

0

dx

x2−4x+ 3 (DS Phˆan k`y)

5.

1

Z

0

xlnxdx. (DS −0,25) 6.

3

Z

2

xdx

4

√

x2−4 (DS

2

4

√

125)

7.

2

Z

0

dx

(x−1)2 (DS Phˆan k`y)

8.

2

Z

−2

xdx

x2−1 (DS Phˆan k`y)

9.

2

Z

0

x3dx

√

4−x2 (DS

16

3 ) Chı’ dˆa˜n. D˘a.tx = sint.

10.

0

Z

−1

e1/x

x3 dx. (DS −

11.

1

Z

0

e1/x

x3 dx. (DS Phˆan k`y)

12.

1

Z

0

dx

p

x(1−x) (DS π) 13.

b

Z

a

dx

p

(x−a)(b−x); a < b. (DS π) 14.

1

Z

0

xln2xdx. (DS 4)

Kha’o s´at su hˆo.i tu cu’a c´ac t´ıch phˆan suy rˆo.ng sau dˆay.

15.

1

Z

0

cos2x

3

√

1−x2dx. (DS Hˆo.i tu.)

16.

1

Z

0

ln(1 +√3x

esinx−1 dx. (DS Hˆo.i tu.) 17.

1

Z

0

dx

e√x−1 (DS Hˆo.i tu.) 18.

1

Z

0

√

xdx

esinx−1 (DS Hˆo.i tu.) 19.

1

Z

0

x2dx

3

p

(1−x2)5 (DS Phˆan k`y)

20.

1

Z

0

x3dx

3

p

21.

1

Z

0

dx

ex−cosx (DS Phˆan k`y) 22.

π/4

Z

0

ln(sin 2x)

5

√

x dx. (DS Hˆo.i tu.)

23.

1

Z

0

lnx

√

xdx. (DS Hˆo.i tu.)

Chı’ dˆa˜n. Su.’ du.ng hˆe th´u.c lim x→0+0x

αlnx= 0 ∀α >0⇒ c´o thˆe’ lˆa´y α=

4 ch˘a’ng ha.n ⇒

|lnx| √

x < x3/4

24.

1

Z

0

sinx

x2 dx. (DS Phˆan k`y)

25.

2

Z

0

dx

√

x−x3 (DS Hˆo.i tu.)

26.

2

Z

1

(x−2)

x2−3x2+ 4dx. (DS Phˆan k`y)

27.

1

Z

0

dx

p

x(ex−e−x) (DS Hˆo.i tu.) 28.

2

Z

0

s

16 +x4

16−x4dx. (DS Hˆo.i tu.)

29.

1

Z

0

√

ex−1

sinx dx. (DS Hˆo.i tu.)

30.

1

Z

0

3

p

ln(1 +x)

T´ıch phˆan h`am nhiˆ`u biˆe e´n

12.1 T´ıch phˆan 2-l´o.p 118

12.1.1 Tru.`o.ng ho p miˆe`n ch˜u nhˆa.t 118 12.1.2 Tru.`o.ng ho..p miˆe`n cong 118 12.1.3 Mˆo.t v`ai ´u.ng du.ng h`ınh ho.c 121

12.2 T´ıch phˆan 3-l´o.p 133

12.2.1 Tru.`o.ng ho p miˆe`n h`ınh hˆo.p 133 12.2.2 Tru.`o.ng ho p miˆe`n cong 134 12.2.3 136 12.2.4 Nhˆa.n x´et chung 136

12.3 T´ıch phˆan d u.`o.ng 144

12.3.1 C´ac di.nh ngh˜ıa co ba’n 144 12.3.2 T´ınh t´ıch phˆan du.`o.ng 146

12.4 T´ıch phˆan m˘a t 158

12.4.3 Cˆong th´u.c Gauss-Ostrogradski 162 12.4.4 Cˆong th´u.c Stokes 162

12.1 T´ıch phˆan 2-l´o.p

12.1.1 Tru.`o.ng ho..p miˆe`n ch˜u nhˆa t Gia’ su.’

D = [a, b]×[c, d] ={(x, y) :a 6x6b, c 6y 6d}

v`a h`am f(x, y) liˆen tu.c miˆe`n D Khi d´o t´ıch phˆan 2-l´o.p cu’a h`am f(x, y) theo miˆ`n ch˜e u nhˆa.t

D ={(x, y) :a6x6 b;c6y6d}

du.o c t´ınh theo cˆong th´u.c

ZZ

D

f(M)dxdy= b

Z

a dx

d

Z

c

f(M)dy; (12.1)

ZZ

D

f(M)dxdy= d

Z

c dy

b

Z

a

f(M)dx, M = (x, y) (12.2)

Trong (12.1): dˆ` u tiˆen t´ınh t´ıch phˆan tronga I(x) theoy xemxl`a h˘a`ng sˆo´, sau d´o t´ıch phˆan kˆe´t qua’ thu du.o c I(x) theo x Dˆo´i v´o.i (12.2) ta c˜ung tiˆe´n h`anh tu.o..ng tu nhu.ng theo th´u tu ngu.o c la.i

12.1.2 Tru.`o.ng ho..p miˆe`n cong Gia’ su.’ h`am f(x, y) liˆen tu.c miˆe`n bi ch˘a.n

trong d´o y=ϕ1(x) l`a biˆen du.´o.i,y =ϕ2(x) l`a biˆen trˆen, ho˘a.c

D ={(x, y) :c6y6d;g1(y)6x6g2(y)}

trong d´o x = g1(y) l`a biˆen tr´ai c`on x = g2(y) l`a biˆen pha’i, o.’ dˆay

ta luˆon gia’ thiˆe´t c´ac h`am ϕ1, ϕ2, g1, g2 dˆ`u liˆen tu.c c´ac khoa’nge

tu.o.ng ´u.ng Khi d´o t´ıch phˆan 2-l´o.p theo miˆ`ne D luˆon luˆon tˆ` n ta.i.o Dˆe’ t´ınh t´ıch phˆan 2-l´o.p ta c´o thˆe’ ´ap du.ng mˆo.t hai phu.o.ng ph´ap sau

1+ Phu.o.ng ph´ap Fubini du a trˆen di.nh l´y Fubini vˆe` viˆe.c du.a t´ıch phˆan 2-l´o.p vˆ` t´ıch phˆan l˘a.p Phu.o.ng ph´ap n`ay cho ph´ep ta du.a t´ıche phˆan 2-l´o.p vˆ` t´ıch phˆan l˘a.p theo hai th´u tu kh´ac nhau:e

ZZ

D

f(M)dxdy = b

Z

a

h ϕZ2(x)

ϕ1(x)

f(M)dyidx = b

Z

a dx

ϕZ2(x)

ϕ1(x)

f(M)dy, (12.3)

ZZ

D

f(M)dxdy = d

Z

c

h gZ2(y)

g1(y)

f(M)dxidy = d

Z

c dy

gZ2(y)

g1(y)

f(M)dx (12.4)

T`u (12.3) v`a (12.4) suy r˘a`ng cˆa n cu’a c´ac t´ıch phˆan biˆe´n thiˆen v`a phu thuˆo c v`ao biˆe´n m`a t´ınh t´ıch phˆan trong, n´o du.o..c xem l`a khˆong dˆo’i Cˆa n cu’a t´ıch phˆan ngo`ai luˆon luˆon l`a h˘a`ng sˆo´.

Nˆe´u cˆong th´u.c (12.3) (tu.o.ng ´u.ng: (12.4)) phˆ` n biˆen du.´o.ia hay phˆ` n biˆen trˆen (tu.o.ng ´a u.ng: phˆ` n biˆen tr´ai hay pha’i) gˆoa ` m t`u mˆo.t sˆo´ phˆ` n v`a mˆo˜i phˆaa ` n c´o phu.o.ng tr`ınh riˆeng th`ı miˆe`nD cˆ` n chia th`anha nh˜u.ng miˆ`n bo.e ’ i c´ac du.`o.ng th˘a’ng song song v´o.i tru.c Oy (tu.o.ng ´

u.ng: song song v´o.i tru.cOx) cho mˆo˜i miˆe`n d´o c´ac phˆa` n biˆen du.´o.i hay trˆen (tu.o.ng ´u.ng: phˆ` n biˆen tr´ai, pha’i) dˆea `u chı’ du.o c biˆe’u diˆe˜n bo.’ i mˆo.t phu.o.ng tr`ınh.

2+ Phu.o.ng ph´ap dˆo’i biˆe´n Ph´ep dˆo’i biˆe´n t´ıch phˆan 2-l´o.p du.o c thu c hiˆe.n theo cˆong th´u.c

ZZ

D

f(M)dxdy=

Z Z

D∗

f[ϕ(u, v), ψ(u, v)]

D(x, y) D(u, v)

trong d´o D∗ l`a miˆ`n biˆe´n thiˆen cu’a to.a dˆo cong (e u, v) tu.o.ng ´u.ng c´ac diˆe’m (x, y) biˆe´n thiˆen D: x = ϕ(u, v), y = ψ(u, v); (u, v)∈D∗, (x, y)∈D; c`on

J = D(x, y) D(u, v) =

∂x ∂u

∂x ∂v ∂y ∂u

∂y ∂v

6= (12.6)

l`a Jacobiˆen cu’a c´ac h`am x=ϕ(u, v),y =ψ(u, v).

To.a dˆo cong thu.`o.ng d`ung ho.n ca’ l`a to.a dˆo cu c (r, ϕ) Ch´ung liˆen hˆe v´o.i to.a dˆo Dˆecac bo.’i c´ac hˆe th´u.c x = rcosϕ, y = rsinϕ, r < +∞, ϕ < 2π T`u (12.6) suy J = r v`a to.a dˆo. cu c (12.5) c´o da.ng

ZZ

D

f(M)dxdy =

ZZ

D∗

f(rcosϕ, rsinϕ)rdrdϕ. (12.7)

K´y hiˆe.u vˆe´ pha’i cu’a (12.7) l`a I(D∗) C´o c´ac tru.`

o.ng ho p cu thˆe’ sau dˆay

(i) Nˆe´u cu c cu’a hˆe to.a dˆo cu c n˘a`m ngo`aiD th`ı

I(D∗) = ϕ2

Z

ϕ1

dϕ rZ2(ϕ)

r1(ϕ)

f(rcosϕ, rsinϕ)rdr. (12.8)

(ii) Nˆe´u cu..c n˘a`m D v`a mˆo˜i tia di t`u cu..c c˘a´t biˆen ∂D khˆong qu´a mˆo.t diˆe’m th`ı

I(D∗) =

2π

Z

0

dϕ r(ϕ)

Z

0

f(rcosϕ, rsinϕ)rdr. (12.9)

(iii) Nˆe´u cu..c n˘a`m trˆen biˆen ∂D cu’a D th`ı

I(D∗) = ϕ2

Z

ϕ1

dϕ r(ϕ)

Z

0

12.1.3 Mˆo t v`ai ´u.ng du ng h`ınh ho c

1+ Diˆe.n t´ıchSD cu’a miˆ`n ph˘a’nge D du.o c t´ınh theo cˆong th´u.c SD =

ZZ

D

dxdy⇒SD =

ZZ

D∗

rdrdϕ. (12.11)

2+ Thˆe’ t´ıch vˆa.t thˆe’ h`ınh tru th˘a’ng d´u.ng c´o d´ay l`a miˆe`nD (thuˆo.c m˘a.t ph˘a’ng Oxy) v`a gi´o.i ha.n ph´ıa trˆen bo’ i m˘ a.t z =f(x, y)>0 du.o..c t´ınh theo cˆong th´u.c

V =

ZZ

D

f(x, y)dxdy. (12.12)

3+ Nˆe´u m˘

a.t (σ) du.o..c cho bo.’i phu.o.ng tr`ınh z = f(x, y) th`ı diˆe.n t´ıch cu’a n´o du.o c biˆe’u diˆe˜n bo.’i t´ıch phˆan 2-l´o.p

Sσ =

ZZ

D(x,y)

q

1 + (f0

x)2+ (fy0)2dxdy, (12.13)

trong d´o D(x, y) l`a h`ınh chiˆe´u vuˆong g´oc cu’a m˘a.t (σ) lˆen m˘a.t ph˘a’ng to.a dˆo.Oxy.

C ´AC V´I DU. V´ı du 1. T´ınh t´ıch phˆan

Z Z

D

xydxdy, D ={(x, y) : 16x62; 16y 62}.

Gia’i. Theo cˆong th´u.c (12.2):

ZZ

D

xydxdy =

2

Z

1

dy

2

Z

1

T´ınh t´ıch phˆan (xemy l`a khˆong dˆo’i) ta c´o I(x) =

2

Z

1

xydx=yx

2 2

= 2y−

2y. Bˆay gi`o t´ınh t´ıch phˆan ngo`ai:

ZZ D xydxdy = Z

2y−1

2y

dy= 4· N V´ı du 2. T´ınh t´ıch phˆan

Z Z

D

xydxdy nˆe´u D du.o c gi´o.i ha.n bo.’i c´ac du.`o.ng cong y=x−4,y2= 2x

Gia’i. B˘a`ng c´ach du ng c´ac du.`o.ng gi˜u.a c´ac giao diˆe’m A(8,4) v`a B(2,−2) cu’a ch´ung, ba.n do.c s˜e thu du.o c miˆe`n lˆa´y t´ıch phˆan D.

Nˆe´u dˆ` u tiˆen lˆa´y t´ıch phˆan theoa x v`a tiˆe´p dˆe´n lˆa´y t´ıch phˆan theo y th`ı t´ıch phˆan theo miˆ`ne D du.o c biˆe’u diˆe˜n bo.’i mˆo.t t´ıch phˆan bˆo.i

I = ZZ D xydxdy = Z −2 ydy y4 Z

y2/2

xdx,

trong d´o doa.n [−2,4] l`a h`ınh chiˆe´u cu’a miˆ`ne D lˆen tru.cOy T`u d´o I =

4

Z

−2

y

hx2

2

y4

y2/2

i

dy = Z −2 y h

(y+ 4)2 −y

4

4

i

dy= 90

Nˆe´u t´ınh t´ıch phˆan theo th´u tu kh´ac: dˆa` u tiˆen theo y, sau d´o theo xth`ı cˆ` n chia miˆea `nD th`anh hai miˆ`n bo.e ’ i du.`o.ng th˘a’ng quaB v`a song song v´o.i tru.c Oy v`a thu du.o..c

I = ZZ D1 + Z Z D2 = Z xdx √ 2x Z

−√2x

ydy+ Z xdx √ 2x Z

x−4

ydy

=

2

Z

0

xdx·0 +

8

Z

2

xhy

2 √ 2x x−4

i

Nhu vˆa.y t´ıch phˆan 2-l´o.p d˜a cho khˆong phu thuˆo.c th´u tu t´ınh t´ıch phˆan Do vˆa.y, cˆa` n cho.n mˆo.t th´u tu t´ıch phˆan dˆe’ khˆong pha’i chia miˆ`n.e N

V´ı du 3. T´ınh t´ıch phˆan

ZZ

D

(y− x)dxdy d´o miˆ`ne D du.o c gi´o.i ha.n bo.’i c´ac du.`o.ng th˘a’ng y = x+ 1, y = x−3, y = −1

3x+ 3, y=−1

3x+

Gia’i. Dˆe’ tr´anh su. ph´u.c ta.p, ta su.’ du.ng ph´ep dˆo’i biˆe´nu=−y−x; v=y+1

3x v`a ´ap du.ng cˆong th´u.c (12.5) Qua ph´ep dˆo’i biˆe´n d˜a cho.n, du.`o.ng th˘a’ng y=x+ biˆe´n th`anh du.`o.ng th˘a’ng u= 1; c`on y=x−3 biˆe´n th`anhu=−3 m˘a.t ph˘a’ngOuv; tu.o.ng tu.., c´ac du.`o.ng th˘a’ng y=−1

3x+

3,y=−

3x+ biˆe´n th`anh c´ac du.`o.ng th˘a’ngv=

3,v = Do d´o miˆ`ne D∗ tro.’ th`anh miˆ`ne D∗ = [−3,1]×

h7

3,5

Dˆ˜ d`ang thˆa´ye r˘a`ng D(x, y)

D(u, v) =−

4 Do d´o theo cˆong th´u.c (12.5):

ZZ

D

(y−x)dxdy =

ZZ

D∗

h1

4u+ 4v

−−

4u+ 4v

i3

4dudv

=

ZZ

D∗

3

4ududv=

5

Z

7/3

dv

4

Z

−3

3

4udu=−8 N

Nhˆa n x´et. Ph´ep dˆo’i biˆe´n t´ıch phˆan hai l´o.p nh˘a`m mu.c d´ıch do.n gia’n h´oa miˆ`n lˆa´y t´ıch phˆan C´o thˆe’ l´e uc d´o h`am du.´o.i dˆa´u t´ıch phˆan tro.’ nˆen ph´u.c ta.p ho.n.

V´ı du 4. T´ınh t´ıch phˆan

ZZ

D

(x2+y2)dxdy, d´o D l`a h`ınh tr`on gi´o.i ha.n bo.’i du.`o.ng tr`on x2+y2 = 2x

c´o da.ng

x=rcosϕ, y=rsinϕ. (12.14) Thˆe´ (12.14) v`ao phu.o.ng tr`ınh du.`o.ng tr`on ta thu du.o..cr2 = 2rcosϕ⇒

r= ho˘a.c r= cosϕ (dˆay l`a phu.o.ng tr`ınh du.`o.ng tr`on to.a dˆo. cu c) Khi d´o

D∗ =n(r, ϕ) :−π

2 6ϕ6 π

2,06r 62 cosϕ

o

T`u d´o thu du.o c

I =

ZZ

D

(x2+y2)dxdy=

ZZ

D∗

r3drdϕ = π/2

Z

−π/2

dϕ

2 cosZ ϕ

0

r3dr

= π/2

Z

−π/2

r4

4

2 cosϕ

0

dϕ= π/2

Z

−π/2

cos4ϕf ϕ= 3π · N

Nhˆa n x´et. Nˆe´u lˆa´y cu..c ta.i tˆam h`ınh tr`on th`ı x−1 = rcosϕ

y=rsinϕ

D∗ =(r, ϕ) : 06r 61,06ϕ62π}

v`a x2+y2 = + 2rcosϕ+r2 nˆen

I =

Z Z

D∗

r(1 + 2rcosϕ+r2)drdϕ

=

2π

Z

0

dϕ

1

Z

0

(r+ 2r2cosϕ+r3)dr= 3π ·

V´ı du 5. T´ınh thˆe’ t´ıch vˆa.t thˆe’T gi´o.i ha.n bo’ i paraboloid z =x2+y2, m˘a.t tru. y=x2 v`a c´ac m˘

Gia’i. H`ınh chiˆe´u cu’a vˆa.t thˆe’T lˆen m˘a.t ph˘a’ngOxy l`a D(x, y) =n(x, y) :−16x61, x2 6y 61o. Do d´o ´ap du.ng (12.12) ta c´o

V(T) =

ZZ

D(x,y)

zdxdy=

ZZ

D(x,y)

(x2+y2)dxdy =

1

Z

−1

dx

1

Z

x2

(x2+y2)dy

=

1

Z

−1

h

x2y+ y

3

3

1

x2

i

dx= 88 105· N

V´ı du 6. T`ım diˆe.n t´ıch m˘a.t cˆa` u b´an k´ınhR v´o.i tˆam ta.i gˆo´c to.a dˆo

Gia’i. Phu.o.ng tr`ınh m˘a.t cˆa` u d˜a cho c´o da.ng x2+y2+z2 =R2. Do d´o phu.o.ng tr`ınh nu.’ a trˆen m˘a.t cˆa` u l`a

z =pR2 −x2−y2.

Do t´ınh dˆo´i x´u.ng nˆen ta chı’ t´ınh diˆe.n t´ıch nu’ a trˆen l` a du’ Ta c´o ds =

q

1 +z0 x

2

+z0 y

2

dxdy = p Rdxdy R2 −x2−y2 ·

Miˆ`n lˆa´y t´ıch phˆane D(x, y) ={(x, y) :x2+y2

6R2} Do d´o

S =

ZZ

D(x,y)

R

p

R2 −x2−y2dxdy=

x=rcosϕ y=rsinϕ J =r

= 2R

2π

Z

0

dϕ R

Z

0

rdr

√

R2−r2

= 4πR

h

− √

R2 −r2R

i

V´ı du 7. T´ınh diˆe.n t´ıch phˆa` n m˘a.t tru. x2 = 2z gi´o.i ha.n bo’ i giao tuyˆe´n cu’a m˘a.t tru d´o v´o.i c´ac m˘a.t ph˘a’ngx−2y= 0, y= 2x,x=

√

2

Gia’i. Dˆ˜ thˆa´y r˘a`ng h`ınh chiˆe´u cu’a phˆae ` n m˘a.t d˜a nˆeu l`a tam gi´ac v´o.i c´ac ca.nh n˘a`m trˆen giao tuyˆe´n cu’a m˘a.t ph˘a’ng Oxy v´o.i c´ac m˘a.t ph˘a’ng d˜a cho

T`u phu.o.ng tr`ınh m˘a.t tru ta c´oz = x

2

2, vˆa.y ∂z

∂x =x, ∂z

∂y = →dS =

√

1 +x2dxdy.

T`u d´o suy r˘a`ng

S =

2√2

Z

0

√

1 +x2dx 2x

Z

x/2

dy =

2√2

Z

0

x

√

1 +x2dx= 13. N

B `AI T ˆA P

T`ım cˆa.n cu’a t´ıch phˆan hai l´o.p

Z Z

D

f(x, y)dxdy theo miˆ`ne D gi´o.i ha.n bo.’i c´ac du.`o.ng d˜a chı’ (Dˆe’ ng˘a´n go.n ta k´y hiˆe.uf(x, y) =f(−)) 1. x= 3, x= 5, 3x−2y+ = 0, 3x−2y+ =

(DS

5

Z

3

dx

3x+4

5

Z

3x+1

5

f(−)dy) 2. x= 0, y = 0, x+y=

(DS

2

Z

0

dx

2−x

Z

0

3. x2+y2 61,x>0, y>0

(DS

1

Z

0

dx √

1−x2

Z

0

f(−)dy) 4. x+y61,x−y61, x>0

(DS

1

Z

0

dx

1−x

Z

x−1

f(−)dy) 5. y>x2, y64−x2.

(DS √

2

Z

−√2

dx

4−x2

Z

x2

f(−)dy) 6. x

2

4 + y2

9 61

(DS

+2

Z

−2

dx

3

√

4−x2

Z

−3

2

√

4−x2

f(−)dy) 7. y=x2,y =√x.

(DS

1

Z

0

dx √

x

Z

x2

f(−)dy) 8. y=x, y= 2x, x+y =

(DS

2

Z

0

dx

2x

Z

x

f(−)dy+

3

Z

2

dx

6−x

Z

x

9. Z dy Z y

f(−)dx (DS

4 Z dx x Z

f(−)dy) 10. Z −1 dx √

1−x2

Z

x+1

f(−)dy (DS

1

Z

0

dy y−1

Z

−

√

1−y2

f(−)dx) 11.

1

Z

0

dx

2−x2

Z

x

f(−)dy (DS

1 Z dy y Z

f dx+

2

Z

1

dy √

2−y

Z f dx) 12. Z dy y Z 1/y

f dx. (DS

1

Z

1/2

dx

2

Z

1/x

f dy+

2 Z dx Z x f dy)

T´ınh c´ac t´ıch phˆan l˘a.p sau 13. Z dx 2x Z x

(x−y+ 1)dy (DS 3) 14. Z −2 dy y Z y3

x2+y2dx. (DS 6π)

15. Z dy y2 Z

(x+ 2y)dx (DS −11,2) 16.

5

Z

0

dx

5−x

Z

0

p

4 +x+ydy. (DS 506 15 ) 17. Z dx Z dy

(x+y)2 (DS

25 24) 18. a Z dx

2Z√ax

−2√ax

(x2+y2)dy (DS 344 105a

4

19.

2π

Z

0

dϕ a

Z

asinϕ

rdr. (DS πa

2

2 )

20∗.

1

Z

0

dx √

1−x2

Z

0

p

1−x2−y2dy. (DS. π

6)

T´ınh c´ac t´ıch phˆan 2-l´o.p theo c´ac h`ınh ch˜u nhˆa.t d˜a chı’ ra. 21.

ZZ

D

(x+y2)dxdy; 26x63, 16 y62 (DS 45 6) 22.

ZZ

D

(x2+y)dxdy; 16x62, 06 y61 (DS 25 6) 23.

ZZ

D

(x2+y2)dxdy; 6x61, 6y 61 (DS 3) 24.

ZZ

D

3y2dxdy

1 +x2 ; 6x61, 6y61 (DS

π 4) 25.

ZZ

D

sin(x+y)dxdy; 0 6x6 π

2, 06y π

2 (DS 2) 26.

ZZ

D

xexydxdy; 06 x61, −16y60 (DS e) 27.

ZZ

D

dxdy

(x−y)2; 6x62, 36y64 (DS ln

4 3)

T´ınh c´ac t´ıch phˆan 2-l´o.p theo miˆ`ne D gi´o.i ha.n c´ac du.`o.ng d˜a chı’

28.

ZZ

D

xydxdy; y = 0, y=x,x= (DS 8) 29.

ZZ

D

xydxdy; y =x2, x=y2. (DS.

30.

ZZ

D

xdxdy;y=x3, x+y= 2, x= (DS 15) 31.

ZZ

D

xdxdy;xy = 6, x+y−7 = (DS 205 6) 32.

ZZ

D

y2xdxdy; x2+y2= 4, x+y−2 = 0. (DS 13

5) 33.

ZZ

D

(x+y)dxdy; 0 6y6π, 06x6 siny. (DS 5π ) 34.

ZZ

D

sin(x+y)dxdy;x=y, x+y = π

2, y= (DS 2) 35.

ZZ

D

e−y2dxdy; D l`a tam gi´ac v´o.i dı’nh O(0,0), B(0,1), A(1,1) (DS −

2e + 2) 36.

ZZ

D

xydxdy;D l`a h`ınh elip 4x2+y2 64 (DS 0)

37.

ZZ

D

x2ydxdy; y= 0, y=√2ax−x2. (DS. 4a

5 ) 38.

ZZ

D

xdxdy

x2+y2; y=x, x= 2, x= 2y (DS

π

2 −2arctg 2) 39.

ZZ

D

√

x+ydxdy; x= 0, y= 0, x+y= (DS 5) 40.

ZZ

D

(x−y)dxdy; y= 2−x2, y = 2x−1. (DS 4

15) 41.

ZZ

D

42.

ZZ

D

xdxdy; x= + siny,x = 0, y= 0, y= 2π (DS 9π ) 43.

ZZ

D

xydxdy; (x−2)2 +y2 = (DS 3) 44.

ZZ

D

dxdy

√

2a−x; D l`a h`ınh tr`on b´an k´ınh a n˘a`m g´oc vuˆong I v`a tiˆe´p x´uc v´o.i c´ac tru.c to.a dˆo (DS.

3a

√

2a) 45.

ZZ

D

ydxdy; x=R(t−sint),y=R(1−cost), 06 t62π (l`a miˆ`ne gi´o.i ha.n bo.’i v`om cu’a xicloid.) (DS.

2πR

3)

Chı’ dˆa˜n.

ZZ

D

ydxdy =

2πR

Z

0

dx y=Zf(x)

0

ydy

Chuyˆe’n sang to.a dˆo cu..c v`a t´ınh t´ıch phˆan to.a dˆo m´o.i 46.

ZZ

D

(x2+y2)dxdy; D :x2+y2 6R2, y>0. (DS. πR

4 ) 47.

ZZ

D

ex2+y2dxdy; D :x2+y2 61, x>0, y>0 (DS π

4(e−1)) 48.

ZZ

D

ex2+y2dxdy; D :x2+y2 6R2 (DS 2π(eR2 −1)) 49.

ZZ

D

p

1−x2 −y2dxdy; D :x2

+y2 6x. (DS π−

4

)

50.

ZZ

D

s

1−x2−y2

1 +x2+y2dxdy; D: x

2+y2 61, x>0,y >0.

(DS π(π−2)

51.

ZZ

D

ln(x2+y2)

x2+y2 dxdy; D : 6x

+y2 6e. (DS 2π)

52.

ZZ

D

(x2+y2)dxdy; D gi´o.i ha.n bo.’i c´ac du.`o.ng tr`on

x2+y2+ 2x−1 = 0, x2+y2+ 2x= (DS 5π )

Chı’ dˆa˜n. D˘a.t x−1 =rcosϕ, y=rsinϕ.

T´ınh thˆe’ t´ıch cu’a vˆa.t thˆe’ gi´o.i ha.n bo.’i c´ac m˘a.t d˜a chı’ ra. 53. x= 0, y = 0, z = 0, x+y+z = (DS

6)

54. x= 0, y = 0, z = 0, x+y= 1, z =x2+y2. (DS.

6) 55. z =x2+y2, y=x2, y = 1, z= 0. (DS. 88

105) 56. z =px2+y2, x2+y2=a2, z = 0. (DS.

3πa

3)

57. z =x2+y2, x2+y2=a2, z = (DS πa

4

2 ) 58. z =x, x2+y2=a2, z = 0. (DS. 4a

3

3 )

59. z = 4−x2 −y2, x=±1, y=±1 (DS 131 3) 60. 2−x−y−2z = 0, y=x2,y =x. (DS 11

120) 61. x2+y2 = 4x,z =x, z = 2x (DS 4π)

T´ınh diˆe.n t´ıch c´ac phˆa` n m˘a.t d˜a chı’ ra.

62. Phˆ` n m˘a.t ph˘a’ng 6a x+ 3y+ 2z = 12 n˘a`m g´oc phˆa` n t´am I (DS 14)

63. Phˆ` n m˘a.t ph˘a’nga x+y+z = 2a n˘a`m m˘a.t tru.x2+y2 =a2.

64. Phˆ` n m˘a.t paraboloida z=x2+y2 n˘a`m m˘a.t tru x2+y2 = (DS π

6(17

√

17−1))

65. Phˆ` n m˘a.t 2a z =x2+y2 n˘a`m m˘a.t tru. x2+y2 = (DS

3(2

√

2−1)π)

66. Phˆ` n m˘a.t n´ona z =px2+y2 n˘a`m m˘a.t tru. x2

+y2 =a2 (DS πa2

√

2)

67. Phˆ` n m˘a.t cˆaa ` ux2+y2+z2 =R2 n˘a`m m˘a.t tru.x2+y2 =Rx.

(DS 2R2(π−2))

68. Phˆ` n m˘a.t n´ona z2 =x2+y2 n˘a`m m˘a.t tru. x2+y2 = 2x (DS

√

2π)

69. Phˆ` n m˘a.t tru.a z2 = 4x n˘a`m g´oc phˆa` n t´am th´u I v`a gi´

o.i ha.n bo.’ i m˘a.t tru.y2 = 4x v`a m˘

a.t ph˘a’ng x= (DS 3(2

√

2−1))

70. Phˆ` n m˘a.t cˆaa ` ux2+y2+z2 =R2 n˘a`m m˘a.t tru.x2+y2 =a2 (a6 R) (DS 4πa(a−√a2−R2))

12.2 T´ıch phˆan 3-l´o.p

12.2.1 Tru.`o.ng ho..p miˆe`n h`ınh hˆo p Gia’ su.’ miˆe`nD ⊂ R3:

D= [a, b]×[c, d]×[e, g] ={(x, y, z) :a6 x6b, c6y6d, e6z 6g}

v`a h`am f(x, y, z) liˆen tu.c trong D Khi d´o t´ıch phˆan 3-l´o.p cu’a h`am f(x, y, z) theo miˆ`ne D du.o..c t´ınh theo cˆong th´u.c

ZZ Z

D

f(x, y, z)dxdydz= b

Z

a

nZd c

hZg

e

f(x, y, z)dz

i

dy

o

dx

= b

Z

a dx

d

Z

c dy

g

Z

e

T`u (12.15) suy c´ac giai doa.n t´ınh t´ıch phˆan 3-l´o.p: (i) Dˆ` u tiˆen t´ınha I(x, y) =

g

Z

e

f(M)dz;

(ii) Tiˆe´p theo t´ınhI(x) = d

Z

c

I(x, y)dy;

(iii) Sau c`ung t´ınh t´ıch phˆan I = b

Z

a

I(x)dx.

Nˆe´u t´ıch phˆan (12.15) du.o c t´ınh theo th´u tu kh´ac th`ı c´ac giai doa.n t´ınh vˆa˜n tu.o.ng tu : dˆa` u tiˆen t´ınh t´ıch phˆan trong, tiˆe´p dˆe´n t´ınh t´ıch phˆan gi˜u.a v`a sau c`ung l`a t´ınh t´ıch phˆan ngo`ai

12.2.2 Tru.`o.ng ho..p miˆe`n cong 1+ Gia’ su.’ h`am f(M) liˆen tu.c miˆe`n bi ch˘a.n

D =(x, y, z) :a6x6b, ϕ1(x)6y6ϕ2(x), g1(x, y)6z 6g2(x, y) .

Khi d´o t´ıch phˆan 3-l´o.p cu’a h`am f(M) theo miˆ`ne D du.o c t´ınh theo cˆong th´u.c

ZZZ

D

f(M)dxdydz = b

Z

a

n ϕZ2(x)

ϕ1(x)

h g2Z(x,y)

g1(x,y)

f(M)dxidyodx (12.16)

ho˘a.c

Z ZZ

D

f(M)dxdydz =

Z Z

D(x,y)

dxdy g2Z(x,y)

g1(x,y)

f(M)dz, (12.17)

thu.`o.ng theo (12.16) t`u t´ıch phˆan trong, tiˆe´p dˆe´n t´ıch phˆan gi˜u.a v`a sau c`ung l`a t´ınh t´ıch phˆan ngo`ai Khi t´ınh t´ıch phˆan 3-l´o.p theo cˆong th´u.c (12.17): dˆ` u tiˆen t´ınh t´ıch phˆan v`a sau d´o c´o thˆe’ t´ınh t´ıcha phˆan 2-l´o.p theo miˆ`ne D(x, y) theo c´ac phu.o.ng ph´ap d˜a c´o 12.1 2+ Phu.o.ng ph´ap dˆo’i biˆe´n Ph´ep dˆo’i biˆe´n t´ıch phˆan 3-l´o.p

du.o c tiˆe´n h`anh theo cˆong th´u.c

Z ZZ

D

f(M)dxdydz =

Z ZZ

D∗

fϕ(u, v, w), ψ(u, v, w), χ(u, v, w)× ×

D(u, v, w)D(x, y, z)dudvdw, (12.18) d´oD∗ l`a miˆ`n biˆe´n thiˆen cu’a to.a dˆo conge u, v, w tu.o.ng ´u.ng khi c´ac diˆe’m (x, y, z) biˆe´n thiˆen D: x = ϕ(u, v, w), y =ψ(u, v, w), z=χ(u, v, w), D(x, y, z)

D(u, v, w) l`a Jacobiˆen cu’a c´ac h`am ϕ, ψ, χ

J = D(x, y, z) D(u, v, w) =

∂ϕ ∂u

∂ϕ ∂v

∂ϕ ∂w ∂ψ

∂u ∂ψ

∂v ∂ψ ∂w ∂χ

∂u ∂χ ∂v

∂χ ∂w

6

= (12.19)

Tru.`o.ng ho p d˘a.c biˆe.t cu’a to.a dˆo cong l`a to.a dˆo tru v`a to.a dˆo cˆa` u (i) Bu.´o.c chuyˆe’n t`u to.a dˆo Dˆec´ac sang to.a dˆo tru (r, ϕ, z) du.o c thu c hiˆe.n theo c´ac hˆe th´u.c x = rcosϕ, y = rsinϕ, z = z; 0 r < +∞, 06 ϕ < 2π, −∞ < z < +∞ T`u (12.19) suy J =r v`a to.a dˆo tru ta c´o

ZZ Z

D

f(M)dxdydz=

ZZ Z

D∗

frcosϕ, rsinϕ, zrdrdϕdz, (12.20)

(ii) Bu.´o.c chuyˆe’n t`u to.a dˆo Dˆec´ac sang to.a dˆo cˆa` u (r, ϕ, θ) du.o..c thu c hiˆe.n theo c´ac hˆe th´u.c x = rsinθcosϕ, y = rsinθsinϕ, z = rcosθ, 0 r < +∞, ϕ < 2π, θ π T`u (12.19) ta c´o J =r2sinθ v`

a to.a dˆo cˆa` u ta c´o

ZZZ

D

f(M)dxdydz =

=

ZZ Z

D∗

frsinθcosϕ, rsinθsinϕ, rcosθr2sinθdrdϕdθ, (12.21) d´o D∗ l`a miˆ`n biˆe´n thiˆen cu’a to.a dˆo cˆae ` u tu.o.ng ´u.ng diˆe’m (x, y, z) biˆe´n thiˆen D.

12.2.3

Thˆe’ t´ıch cu’a vˆa.t thˆe’ cho´an hˆe´t miˆe`n D ⊂ R3

du.o c t´ınh theo cˆong th´u.c

VD =

Z ZZ

D

dxdydz. (12.22)

12.2.4 Nhˆa n x´et chung

B˘a`ng c´ach thay dˆo’i th´u tu. t´ınh t´ıch phˆan t´ıch phˆan 3-l´o.p ta s˜e thu du.o c c´ac cˆong th´u.c tu.o.ng tu nhu cˆong th´u.c (12.16) dˆe’ t´ınh t´ıch phˆan Viˆe.c t`ım cˆa.n cho t´ıch phˆan do.n thˆong thu.`o.ng chuyˆe’n t´ıch phˆan 3-l´o.p vˆ` t´ıch phˆan l˘a.p du.o c thu c hiˆe.n nhu dˆo´i v´o.i tru.`o.ng ho pe t´ıch phˆan 2-l´o.p

C ´AC V´I DU. V´ı du 1. T´ınh t´ıch phˆan l˘a.p

I =

1

Z

−1

dx

1

Z

x2

dy

2

Z

0

Gia’i. Ta t´ınh liˆen tiˆe´p ba t´ıch phˆan x´ac di.nh thˆong thu.`o.ng b˘a´t dˆ` u t`a u t´ıch phˆan

I(x, y) =

2

Z

0

(4 +z)dz = 4z20+ z

2

2

2 = 10;

I(x) =

1

Z

x2

I(x, y)dy= 10

1

Z

x2

dy= 10(1−x2); I =

1

Z

−1

I(x)dx=

1

Z

−1

10(1−x2)dx= 40 · N V´ı du 2. T´ınh t´ıch phˆan

I =

ZZ Z

D

(x+y+z)dxdydz,

trong d´o miˆ`ne D du.o c gi´o.i ha.n bo.’i c´ac m˘a.t ph˘a’ng to.a dˆo v`a m˘a.t ph˘a’ng x+y+z =

Gia’i. Miˆ`ne D d˜a cho l`a mˆo.t t´u diˆe.n c´o h`ınh chiˆe´u vuˆong g´oc trˆen m˘a.t ph˘a’ng Oxy l`a tam gi´ac gi´o.i ha.n bo’ i c´ ac du.`o.ng th˘a’ng x = 0, y = 0, x+y = R˜o r`ang l`a x biˆe´n thiˆen t`u dˆe´n (doa.n [0,1] l`a h`ınh chiˆe´u cu’a D lˆen tru.c Ox) Khi cˆo´ di.nh x, 0 x th`ıy biˆe´n thiˆen t`u dˆe´n 1−x Nˆe´u cˆo´ di.nh ca’xv`ay(0 6x61, 06y61−x) th`ı diˆe’m (x, y, z) biˆe´n thiˆen theo du.`o.ng th˘a’ng d´u.ng t`u m˘a.t ph˘a’ng z = dˆe´n m˘a.t ph˘a’ng x+y +z = 1, t´u.c l`a z biˆe´n thiˆen t`u dˆe´n 1−x−y Theo cˆong th´u.c (12.16) ta c´o

I =

1

Z

0

dx

1−x

Z

0

dy

1−Zx−y

0

Dˆ˜ d`ang thˆa´y r˘a`nge I =

1

Z

0

dx

1−x

Z

0

xz+yz+z

2

2

i1−x−y

0

dy

=

1

Z

0

nh

y−yx2−xy2− y

3

3

i1−x

0

o

dx

=

1

Z

0

(2−3x+x3)dx= 8· N V´ı du 3. T´ınh I =

ZZZ

D

dxdydz

(x+y+z)3, d´o miˆ`ne D du.o c gi´o.i

ha.n bo.’i c´ac m˘a.t ph˘a’ngx+z = 3, y= 2, x= 0, y= 0, z =

Gia’i. Miˆ`ne D d˜a cho l`a mˆo.t h`ınh l˘ang tru c´o h`ınh chiˆe´u vuˆong g´oc lˆen m˘a.t ph˘a’ng Oxy l`a h`ınh ch˜u nhˆa.t D(x, y) = (x, y) :

x 3,0 y V´o.i diˆe’m M(x, y) cˆo´ di.nh thuˆo.c D(x, y) diˆe’m (x, y, z) ∈ D biˆe´n thiˆen trˆen du.`o.ng th˘a’ng d´u.ng t`u m˘a.t ph˘a’ng Oxy (z = 0) dˆe´n m˘a.t ph˘a’ng x+z = 3, t´u.c l`a z biˆe´n thiˆen t`u dˆe´n 3−x: 06z 63−x T`u d´o theo (12.17) ta c´o

ZZZ

D

f(M)dxdydz =

ZZ

D(x,y)

dxdy z=3Z−x

z=0

(x+y+z+ 1)−3dz

=

ZZ

D(x,y)

h(x+y+z+ 1)−2

−2

3−x

0

i

dxdy =· · ·= ln 2−1

8 · N

V´ı du 4. T´ınh t´ıch phˆan

ZZZ

D

(x2+y2+z2)dxdydz, d´o miˆ`ne D du.o c gi´o.i ha.n bo.’i m˘a.t 3(x2+y2) +z2 = 3a2

Gia’i. Phu.o.ng tr`ınh m˘a.t biˆen cu’aD c´o thˆe’ viˆe´t du.´o.i da.ng x2

a2 +

y2

b2 +

z2

(a

√

D´o l`a m˘a.t elipxoid tr`on xoay, t´u.c l`a D l`a h`ınh elipxoid tr`on xoay H`ınh chiˆe´u vuˆong g´ocD(x, y) cu’aD lˆen m˘a.t ph˘a’ngOxy l`a h`ınh tr`on x2 +y2 6 a2 Do d´o ´

ap du.ng c´ach lˆa.p luˆa.n nhu c´ac v´ı du v`a ta thˆa´y r˘a`ng diˆe’m M(x, y) ∈ D(x, y) du.o c cˆo´ di.nh th`ı diˆe’m (x, y, z) cu’a miˆ`ne D biˆe´n thiˆen trˆen du.`o.ng th˘a’ng d´u.ng M(x, y) t`u m˘a.t biˆen du.´o.i cu’a D

z =−p3(a2−x2−y2)

dˆe´n m˘a.t biˆen trˆen

z = +p3(a2−x2−y2).

T`u d´o theo (12.17) ta c´o

I =

Z Z

D(x,y)

dxdy

+

√

3(a2−x2−y2)

Z

−

√

3(a2−x2−y2)

(x2+y2+z2)dz

= 2a2

√

3

Z Z

x2+y26a2

p

a2−x2−y2dxdy=|chuyˆe’n sang to.a dˆo cu c|

= 2a2

√

3

Z Z

r6a

√

a2−r2rdrdϕ=a2√

3

2π

Z

0

dϕ a

Z

0

(a2 −r2)1/2rdr = 4πa

5

√

3 · N

V´ı du 5. T´ınh thˆe’ t´ıch cu’a vˆa.t thˆe’ gi´o.i ha.n bo.’i c´ac m˘a.t ph˘a’ng x+y+z = 4, x= 3, y= 2, x= 0, y = 0, z =

x+y = Do d´o ´ap du.ng (12.17) ta c´o VD = ZZZ D dxdydz = ZZ

D(x,y)

dxdy

4−Zx−y

0

dz =

ZZ

D(x,y)

(4−x−y)dxdy = Z dy Z

(4−x−y)dx+

2

Z

1

dy

4−y

Z

0

(4−x−y)dx =

1

Z

0

nh

(4−y)x− x

2

2

i3 o dy+ Z nh

(4−y)x− x

2

2

i4−y

0 o dy = Z 15

2 −3y

dy+

2

Z

1

(4−y)2dy= 55 · N V´ı du 6. T´ınh t´ıch phˆan

I =

ZZ Z

D

zpx2+y2dxdydz,

trong d´o miˆ`ne D gi´o.i ha.n bo.’i m˘a.t ph˘a’ng y= 0, z = 0, z =a v`a m˘a.t tru x2+y2 = 2x (x>0,y >0,a >0)

Gia’i. Chuyˆe’n sang to.a dˆo tru ta thˆa´y phu.o.ng tr`ınh m˘a.t tru.x2 + y2 = 2x

trong to.a dˆo tru c´o da.ngr= cosϕ, 06ϕ6 π

2 (h˜ay v˜e h`ınh !) Do d´o theo cˆong th´u.c (12.20) ta c´o

I = π/2

Z

0

dϕ

2 cosZ ϕ

0

r2dr a

Z

0

zdz = a

2 π/2 Z dϕ

2 cosZ ϕ

0

r2dr

= 4a π/2 Z

cos3ϕdϕ= 9a

2

. N

V´ı du 7. T´ınh t´ıch phˆan I =

ZZZ

D

nˆe´u miˆ`ne D l`a nu.’ a trˆen cu’a h`ınh cˆ` ua x2+y2+z2 6R2, z >0

Gia’i. Chuyˆe’n sang to.a dˆo cˆa` u, miˆe`n biˆe´n thiˆen D∗ cu’a c´ac to.a dˆo. cˆ` u tu.o.ng ´a u.ng diˆe’m (x, y, z) biˆe´n thiˆen D l`a c´o da.ng

D∗ : 06ϕ <2π, 06 θ6 π

2, 06r6R. T`u d´o

I =

ZZZ

D∗

r2sin2θ·r2sinθdrdϕdθ=

2π Z dϕ π/2 Z

sin3θdθ R

Z

0

r4dr

= 15πR

5

. N

B `AI T ˆA P

T´ınh c´ac t´ıch phˆan l˘a.p sau

1. Z dx √ x Z ydy

2Z−2x

1−x

dz. (DS 12) 2. a Z ydy h Z dx a−y

Z

0

dz. (DS a

3 h ) 3. Z dy Z √

2y−y2

xdx

3

Z

0

z2dz. (DS 30)

4.

1

Z

0

dx

1−x

Z

0

dy

1−Zx−y

0

dz

(1 +x+y+z)3 (DS

ln 2 − 16) 5. c Z dz b Z dy a Z

(x2+y2+z2)dx (DS abc (a

6. a

Z

0

dx a−x

Z

0

dy a−Zx−y

0

(x2+y2 +z2)dz (DS a

5

20)

T´ınh c´ac t´ıch phˆan 3-l´o.p theo miˆ`ne D gi´o.i ha.n bo.’i c´ac m˘a.t d˜a chı’

7.

ZZZ

D

(x+y−z)dxdydz; x=−1, x= 1; y= 0, y= 1; z = 0, z = (DS −2)

8.

ZZZ

D

xydxdydz;x= 1, x= 2;y =−2,y=−1;z = 0, z = (DS −8

9) 9.

ZZZ

D

dxdydz

(x+y+z)2;x= 1, x = 2; y= 1, y= 2; z = 1, z =

(DS 2ln

128 125) 10.

ZZZ

D

(x+ 2y+ 3z+ 4)dxdydz;x= 0, x= 3; y = 0, y= 2; z = 0, z = (DS 54)

11.

ZZZ

D

zdxdydz; x= 0, y= 0, z = 0; x+y+z = (DS 24) 12.

ZZZ

D

xdxdydz;x= y= 0, z = 0, y= 1; x+z = (DS 6) 13.

ZZZ

D

yzdxdydz; x2+y2 +z2 = 1, z >0. (DS 0)

14.

ZZZ

D

xydxdydz;x2 +y2= 1, z = 0, z= (x>0,y>0).

(DS 8) 15.

ZZZ

D

(x>0, y>0,z >0) (DS 48) 16.

ZZZ

D

p

x2+y2dxdydz; x2

+y2 =z2,z = 0,z = (DS π/6)

17.

ZZZ

D

(x2+y2+z2)dxdydz;x= 0, x=a, y= 0, y =b, z = 0, z =c. (DS abc

3 (a

2+b2+c2))

18.

ZZZ

D

ydxdydz; y=

√

x2+z2, y=h,h >0. (DS. πh

4 ) T´ınh c´ac t´ıch phˆan 3-l´o.p sau b˘a`ng phu.o.ng ph´ap dˆo’i biˆe´n 19.

ZZZ

D

(x2+y2+z2)dxdydz;x2+y2+z2 6R2. (DS. 4πR

5 ) 20.

ZZZ

D

(x2+y2)dxdydz; z =x2+y2, z = (DS π 6) 21.

ZZZ

D

p

x2+y2+z2dxdydz; x2

+y2+z2 6R2 (DS πR4)

22.

ZZZ

D

zpx2+y2dxdydz; x2+y2 = 2x, y= 0, z= 0, z = 3.

(DS 8) 23.

ZZZ

D

zdxdydz; x2+y2+z2 6R2, x>0, y>0,z >0

(DS πR

4

16 ) 24.

ZZZ

D

(x2−y2)dxdydz; x2+y2 = 2z, z = 2. (DS. 16π

3 ) 25.

ZZZ

D

T´ınh thˆe’ t´ıch cu’a c´ac vˆa.t thˆe’ gi´o.i ha.n bo.’i c´ac m˘a.t d˜a chı’ 26. x= 0, y = 0, z = 0, x+ 2y+z−6 = (DS 36)

27. 2x+ 3y+ 4z = 12; x= 0, y= 0, z = (DS 12) 28. x

a + y b +

z

c = 1, x= 0, y= 0, z = (DS abc

6 ) 29. ax=y2+z2, x=a. (DS. πa

3

2 ) 30. 2z =x2+y2, z = 2. (DS 4π)

31. z =x2+y2, x2+y2+z2 = (DS π 6[8

√

2−7]) 32. z =px2+y2, z=x2

+y2 (DS π 6) 33. x2+y2−z = 1, z = 0. (DS. π

2) 34. 2z =x2+y2, y+z = 4. (DS. 81π

4 ) 35. x

2

a2 +

y2 b2 +

z2

c2 = (DS

4 3πabc)

12.3 T´ıch phˆan du.`o.ng

12.3.1 C´ac di.nh ngh˜ıa co ba’n

Gia’ su.’ h`am f(M), P(M) v`a Q(M),M = (x, y) liˆen tu.c ta.i mo.i diˆe’m cu’a du.`o.ng cong du.o cL=L(A, B) v´o.i diˆe’m dˆ` ua Av`a diˆe’m cuˆo´iB Chia mˆo.t c´ach t`uy ´yL(A, B) th`anh n cung nho’ v´o.i dˆo d`ai tu.o.ng ´u.ng l`a ∆s0, ∆s1, ∆s2, ,∆sn−1 D˘a.t d= max

06i6n−1(∆si) Trong mˆo˜i cung

nho’, lˆa´y mˆo.t c´ach t`uy ´y diˆe’m N0, N1, , Nn−1 t´ınh gi´a tri f(Ni),

P(Ni) v`a Q(Ni) ta.i diˆe’mNi d´o

Phu.o.ng ph´ap I.Lˆa´y gi´a tri.f(Ni) nhˆan v´o.i dˆo d`ai cung ∆si tu.o.ng ´

u.ng v`a lˆa.p tˆo’ng t´ıch phˆan σ1 =

n−1

X

i=0

f(Ni)∆si. (*)

Phu.o.ng ph´ap II. Kh´ac v´o.i c´ach lˆa.p tˆo’ng t´ıch phˆan (∗), phu.o.ng ph´ap n`ay ta lˆa´y gi´a tri. P(Ni), Q(Ni) nhˆan khˆong pha’i v´o.i dˆo d`ai cu’a c´ac cung nho’ m`a l`a nhˆan v´o.i h`ınh chiˆe´u vuˆong g´oc cu’a c´ac cung nho’ d´o trˆen c´ac tru.c to.a dˆo., t´u.c l`a lˆa.p tˆo’ng

σx = n−1

X

i=0

P(Ni)∆xi; ∆xi =proOx∆si,

σy = n−1

X

i=0

Q(Ni)∆yi; ∆yi =proOy∆si

Mˆo˜i c´ach lˆa.p tˆo’ng t´ıch phˆan trˆen dˆay s˜e dˆa˜n dˆe´n mˆo.t kiˆe’u t´ıch phˆan du.`o.ng

D- i.nh ngh˜ıa 12.3.1. Nˆe´u tˆ` n ta.i gi´o.i ha.n h˜u.u ha.n limo

d→0σ1 khˆong phu

thuˆo.c v`ao ph´ep phˆan hoa.ch du.`o.ng cong L th`anh c´ac cung nho’ v`a khˆong phu thuˆo.c v`ao viˆe.c cho.n c´ac diˆe’m trung gian Ni trˆen mˆo˜i cung nho’ th`ı gi´o.i ha.n d´o du.o c go.i l`a t´ıch phˆan du.`o.ng theo dˆo d`ai (hay t´ıch phˆan du.`o.ng kiˆe’u I) cu’a h`am f(x, y) theo du.`o.ng cong L = L(A, B) K´y hiˆe.u:

Z

L

f(x, y)ds. (12.23)

D- i.nh ngh˜ıa 12.3.2. Ph´at biˆe’u tu.o.ng tu nhu di.nh ngh˜ıa 12.3.1:

1+ lim

d→0σx = limd→0

n−1

X

i=0

P(Ni)∆xi =

Z

L(A,B)

P(x, y)dx

go.i l`a t´ıch phˆan du.`o.ng theo ho`anh dˆo (nˆe´u (12.24) tˆo`n ta.i h˜u.u ha.n)

2+ lim

d→0σy = limd→0

n−1

X

i=0

Q(Ni)∆yi=

Z

L(A,B)

Q(x, y)dy

(12.25) go.i l`a t´ıch phˆan du.`o.ng theo tung dˆo (nˆe´u (12.25) tˆo` n ta.i h˜u.u ha.n)

Thˆong thu.`o.ng ngu.`o.i ta lˆa.p tˆo’ng t´ıch phˆan da.ng

Σ = n−1

X

i=0

P(Ni)∆xi+ n−1

X

o=0

Q(Ni)∆yi

v`a nˆe´u∃ lim

d→0Σ th`ı gi´o.i ha.n d´o du.o c go.i l`a t´ıch phˆan du.`o.ng theo to.a

dˆo da.ng tˆo’ng qu´at:

Z

L(A,B)

P(x, y)dx+Q(x, y)dy. (12.26)

D- i.nh l´y. Nˆe´u c´ac h`am f(x, y), P(x, y), Q(x, y) liˆen tu c theo du.`o.ng cong L(A, B) = L th`ı c´ac t´ıch phˆan du.`o.ng (12.23) - (12.26) tˆ` n ta.io h˜u.u ha n.

T`u di.nh ngh˜ıa 12.3.1 v`a kh´ai niˆe.m dˆo d`ai cung (khˆong phu thuˆo.c hu.´o.ng cu’a cung) v`a di.nh ngh˜ıa 12.3.2 v`a t´ınh chˆa´t cu’a h`ınh chiˆe´u cu’a cung (h`ınh chiˆe´u dˆo’i dˆa´u dˆo’i hu.´o.ng cu’a cung) suy t´ınh chˆa´t quan tro.ng cu’a t´ıch phˆan du.`o.ng: t´ıch phˆan du.`o.ng theo dˆo d`ai khˆong phu thuˆo c v`ao hu.´o.ng cu’a du.`o.ng cong; t´ıch phˆan du.`o.ng theo to a dˆo. dˆo’i dˆa´u dˆo’i hu.´o.ng du.`o.ng cong.

12.3.2 T´ınh t´ıch phˆan du.`o.ng

tr`ınh cu’a du.`o.ng lˆa´y t´ıch phˆanL =L(A, B) ta biˆe´n dˆo’i biˆe’u th´u.c du.´o.i dˆa´u t´ıch phˆan du.`o.ng th`anh biˆe’u th´u.c mˆo.t biˆe´n m`a gi´a tri cu’a biˆe´n d´o ta.i diˆe’m dˆa` u A v`a diˆe’m cuˆo´i B s˜e l`a cˆa.n cu’a t´ıch phˆan x´ac di.nh thu du.o c

1+ Nˆe´uL(A, B) du.o..c cho bo.’i c´ac phu.o.ng tr`ınh tham sˆo´x=ϕ(t), y=ψ(t),t∈[a, b] (trong d´o ϕ,ψ kha’ vi liˆen tu.c v`aϕ02+ψ02 >0) th`ı

ds=

q

ϕ02

+ψ02

dt

Z

L(A,B)

f(x, y)ds= b

Z

a

f[ϕ(t), ψ(t)]

q

ϕ02

+ψ02dt (12.27)

v`a

Z

L(A,B)

P(x, y)dx+Q(x, y)dy=

= b

Z

a

P ϕ(t), ψ(t)ϕ0(t) +Q ϕ(t), ψ(t)ψ0(t)dt. (12.28)

2+ Nˆe´u L(A, B

) du.o c cho bo.’i phu.o.ng tr`ınh y = g(x), x ∈ [a, b] (trong d´o g(x) kha’ vi liˆen tu.c trˆen [a, b]) th`ı

ds=

q

1 +g02

(x)dx

Z

L(A,B)

f(x, y)ds= b

Z

a

f[x, g(x)]

q

1 +g02

(x)dx (12.29)

v`a

Z

L(A,B)

P dx+Qdy= b

Z

a

3+ Nˆe´uL(A, B) du.o..c cho du.´o.i da.ng to.a dˆo cu cρ=ρ(ϕ)α 6ϕ6

β th`ı

ds=

q

ρ2+ρ0 ϕ

2

dϕ

Z

L(A,B)

f(x, y)ds= β

Z

α

f[ρcosϕ, ρsinϕ]

q

ρ2+ρ02

dϕ. (12.31)

4+ T´ıch phˆan du.`

o.ng theo to.a dˆo c´o thˆe’ t´ınh nh`o cˆong th´u.c Green Nˆe´u P(x, y), Q(x, y) v`a c´ac da.o h`am riˆeng ∂Q

∂x, ∂P

∂y c`ung liˆen tu.c miˆ`ne D gi´o.i ha.n bo’ i du.` o.ng cong khˆong tu. c˘a´t tro.n t`u.ng kh´uc

L=∂D th`ı

I

L+

P dx+Qdy =

Z Z

D

∂Q

∂x − ∂P

∂y

dxdy. (12.32)

Cˆong th´u.c (12.32) go.i l`a cˆong th´u.c Green, d´o

I

L+

l`a t´ıch phˆan theo du.`o.ng cong k´ın c´o hu.´o.ng du.o.ngL+.

Hˆe qua’. Diˆe.n t´ıch miˆe`n D gi´o.i ha n bo’ i du.`. o.ng cong L du.o..c t´ınh theo cˆong th´u.c

SD =

I

L

xdy−ydx. (12.33)

5+ Nhˆa

n x´et vˆ` t´ıch phˆe an du.`o.ng khˆong gian. Gia’ su.’ L =

L(A, B) l`a du.`o.ng cong khˆong gian; f, P, Q, R l`a nh˜u.ng h`am ba biˆe´n liˆen tu.c trˆenL Khi d´o tu.o.ng tu nhu tru.`o.ng ho p du.`o.ng cong ph˘a’ng ta c´o thˆe’ di.nh ngh˜ıa t´ıch phˆan du.`o.ng theo dˆo d`ai

I

L(A,B)

f(x, y, z)dsv`a t´ıch phˆan du.`o.ng theo to.a dˆo.

Z

L

P(x, y, z)dx,

Z

L

Q(x, y, z)dy,

Z

L

v`a

Z

L

P dx+Qdy+Rdz.

Vˆ` thu c chˆa´t k˜y thuˆa.t t´ınh c´ac t´ıch phˆan n`ay khˆong kh´ac biˆe.t g`ıe so v´o.i tru.`o.ng ho..p du.`o.ng cong ph˘a’ng

C ´AC V´I DU. V´ı du 1. T´ınh t´ıch phˆan du.`o.ng

I

L x

yds, d´o L l`a cung parabˆon y2 = 2x t`u diˆe’m (1,

√

2) dˆe´n diˆe’m (2,2)

Gia’i. Ta t`ım vi phˆan dˆo d`ai cung Ta c´o y=

√

2x, y0= √1

2x, ds =

q

1 +y02

dx=

r

1 + 2xdx=

√

1 + 2x

√

2x dx. Tu d´o suy

I

L x yds=

2

Z

1

x

√

2x ·

√

1 + 2x

√

2x dx= 6[5

√

5−3

√

3] N

V´ı du 2. T´ınh dˆo d`ai cu’a du.`o.ng astroid x = acos3t, y = asin3

t, t∈[0,2π]

Gia’i. Ta ´ap du.ng cˆong th´u.c: dˆo d`ai (L) =

I

L

ds Trong tru.`o.ng ho p n`ay ta c´o

x0=−3acos2tsint, y0= 3asin2tcost, ds= 3a

2 sin 2tdt V`ı du.`o.ng cong dˆo´i x´u.ng v´o.i c´ac tru.c to.a dˆo nˆen

dˆo d`ai(L) = π/2

Z

0

3a

2 sin 2tdt= 6a

h−cos 2t

iπ/2

V´ı du 3. T´ınh

I

L

(x−y)ds, d´o L:x2+y2 = 2ax

Gia’i. Chuyˆe’n sang to.a dˆo cu c x =rcosϕ, y =rsinϕ Trong to.a dˆo cu..c phu.o.ng tr`ınh du.`o.ng tr`on c´o da.ng r = 2acosϕ,−π

2 ϕ6 π Vi phˆan dˆo d`ai cung

ds =

q

r2+r0 ϕ

2

dϕ=

q

4a2cos2ϕ+ 4a2sin2

ϕdϕ= 2adϕ Do d´o

I =

I

L

(x−y)ds= π/2

Z

−π/2

(2acosϕ) cosϕ−(2asinϕ) sinϕ2adϕ = 4a2

π/2

Z

−π/2

cos2ϕdϕ= 2πa2. N

V´ı du 4. T´ınh t´ıch phˆan

I

L

(3x2+y)dx+ (x−2y2)dy, d´o L l`a biˆen cu’a h`ınh tam gi´ac v´o.i dı’nh A(0,0),B(1,0),C(0,1)

Gia’i. Theo t´ınh chˆa´t cu’a t´ıch phˆan du.`o.ng ta c´o

I

L =

I

AB +

I

BC +

I

CA .

a) Trˆen ca.nh AB ta c´oy= ⇒dy = Do d´o

I

AB =

1

Z

0

3x2dx=

b) Trˆen ca.nhBC ta c´ox+y= 1⇒y=−x+ 1, dy=−dx Do d´o

I

BC =

0

Z

1

[3x2+ (1−x)−x+ 2(1−x2)]dx=−5

c) Trˆen ca.nh CA ta c´o x= ⇒dx= v`a d´o

I

CA =−

0

Z

1

2y2dy= 3· Nhu vˆa.y

I

L

= 1−

3 +

3 = N

V´ı du 5. T´ınh t´ıch phˆan

I

L

(x+y)dx−(x−y)dy, d´oLl`a du.`o.ng elip x

2

a2 +

y2

b2 = c´o di.nh hu.´o.ng du.o.ng

Gia’i. 1+Ta c´o thˆ

e’ t´ınh tru c tiˆe´p t´ıch phˆan d˜a cho b˘a`ng c´ac phu.o.ng ph´ap d˜a nˆeu (ch˘a’ng ha.n b˘a`ng c´ach tham sˆo´ h´oa phu.o.ng tr`ınh elip).

2+ Nhu.ng do.n gia’n ho.n ca’ l`a su.

’ du.ng cˆong th´u.c Green Ta c´o P =x+y, Q=−(x−y)⇒ ∂Q

∂x − ∂P

∂y =−2 Do d´o theo cˆong th´u.c Green ta c´o

I

L =

Z Z

x2

a2+

y2 b261

(−2)dxdy=−2πab, v`ı diˆe.n t´ıch h`ınh elip b˘a`ng πab. N

V´ı du 6. T´ınh t´ıch phˆan

I

L

2(x2+y2)dx+x(4y+ 3)dy, d´o Ll`a du.`o.ng gˆa´p kh´ucABC v´o.i dı’nh A(0,0),B(1,1) v`a C(0,2)

Gia’i. Nˆe´u ta nˆo´i A v´o.i C th`ı thu du.o c du.`o.ng gˆa´p kh´uc k´ın L∗ gi´o.i ha.n ∆ABC Trˆen ca.nh CA ta c´o x= nˆendx= v`a t`u d´o

I

CA

Do d´o

I

L +

I

CA =

I

L∗

⇒

I

L =

I

L∗

.

´

Ap du.ng cˆong th´u.c Green ta c´o

I

L =

ZZ

∆ABC

[(4y+ 3)−4y]dxdy=

ZZ

∆ABC dxdy = 3S∆ABC = N

B `AI T ˆA P

T´ınh c´ac t´ıch phˆan du.`o.ng theo dˆo d`ai sau dˆay 1.

I

C

(x+y)ds, C l`a doa.n th˘a’ng nˆo´i A(9,6) v´o.i B(1,2) (DS 36

√

5)

2.

I

C

xyds,C l`a biˆen h`ınh vuˆong |x|+|y|=a, a >0 (DS 0) 3.

I

C

(x+y)ds, C l`a biˆen cu’a tam gi´ac dı’nh A(1,0),B(0,1),C(0,0) (DS +

√

2) 4.

I

C ds

x−y,C l`a doa.n th˘a’ng nˆo´i A(0,2) v´o.iB(4,0) (DS

√

5 ln 2)

5.

I

C

p

x2+y2ds, C l`a du.`o.ng tr`on x2

+y2 =ax. (DS 2a2)

6.

I

C

(x2+y2)nds,C l`a du.`o.ng tr`on x2+y2 =a2. (DS 2πa2n+1)

7.

I

C e

√

x2+y2

(r, ϕ) : 06r6 a,06ϕ6 π

4

(DS 2(ea−1) + πae a

4 ) 8.

I

C

xyds,C l`a mˆo.t phˆa` n tu elip n˘a`m g´oc phˆa` n tu I (DS ab

3 ·

a2+ab+b2 a+b )

Chı’ dˆa˜n. Su.’ du.ng phu.o.ng tr`ınh tham sˆo´ cu’a du.`o.ng elip: x = acost,y=bsint.

9.

I

C

ds

p

x2 +y2+ 4,C l`a doa.n th˘a’ng nˆo´i diˆe’m O(0,0) vo.iA(1,2)

(DS ln

√

5 + ) 10.

I

C

(x2+y2+z2)ds, C l`a cung du.`o.ng cong x=acost, y=asint, z=bt; 06t62π, a >0, b >0

(DS 2π

√

a2+b2(3a2

+ 4π2b2)) 11.

I

C

x2ds, C l`a du.`o.ng tr`on

  

x2+y2+z2 =a2

x+y+z =

(DS 2πa

3

3 )

Chı’ dˆa˜n. Ch´u.ng to’ r˘a`ng

I

C

x2ds=

I

C

y2ds=

I

C

z2ds v`a t`u d´o suy

I =

I

C

12.

I

C

(x+y)ds, C l`a mˆo.t phˆa` n tu du.`o.ng tr`on

  

x2+y2+z2 =R2

y=x n˘a`m g´oc phˆa` n t´am I (DS.R2

√

2) 13. T´ınh

I

C

xyzds, C l`a mˆo.t phˆa` n tu du.`o.ng tr`on

  

x2+y2+z2 =R2 x2+y2 = R

2

4 n˘a`m g´oc phˆa` n t´am I

T´ınh c´ac t´ıch phˆan du.`o.ng theo to.a dˆo sau dˆay 14.

I

C

y2dx+x2dy,C l`a du.`o.ng t`u diˆe’m (0,0) dˆe´n diˆe’m (1,1): 1) C l`a doa.n th˘a’ng.

2) C l`a cung parabol y=x2.

3) C l`a cung parabol y=√x. (DS 1)

3; 2) 10 ; 3)

7 10) 15.

I

C

y2dx−x2dy, C l`a du.`o.ng tr`on b´an k´ınh R = v`a c´o hu.´o.ng ngu.o c chiˆe`u kim dˆo` ng hˆo` v`a:

1) v´o.i tˆam ta.i gˆo´c to.a dˆo 2) v´o.i tˆam ta.i diˆe’m (1,1) (DS 1) 0; 2) −4π) 16.

I

C

v`a (1,2) (DS 2) 17.

I

C

cosydx−sinxdy,C l`a doa.n th˘a’ng t`u diˆe’m (2,−2) dˆe´n diˆe’m (−2,2) (DS −2 sin 2)

18.

I

C

(x2+y2)dx+ (x2−y2)dy,C l`a du.`o.ng cong y= 1− |1−x|, 06x62 (DS

3) 19.

I

C

(x+y)dx+ (x−y)dy,C l`a elip c´o hu.´o.ng du.o.ng x

2

a2 +

y2

b2 =

(DS 0) 20.

I

C

(2a−y)dx+xdy, C l`a mˆo.t v`om cuˆo´n cu’a du.`o.ng xicloid x=a(t−sint),y=a(1−cost), 06t62π (DS −2πa2)

21.

I

C

dx+dy

|z|+|y|, C l`a biˆen c´o hu.´o.ng du.o.ng cu’a h`ınh vuˆong v´o.i dı’nh

ta.i diˆe’m A(1,0), B(0,1), C(−1,0) v`aD(0,−1) (DS 0) 22.

I

C

(x2−y2)dx+ (x2 +y2)dy,C l`a elip c´o hu.´o.ng du.o.ng x2

a2 +

y2

b2 = (DS 0)

23.

I

C

(x2+y2)dx+xydy, C l`a cung cu’a du.`o.ng y=ex t`u diˆe’m (0,1) dˆe´n diˆe’m (1, e) (DS 3e

2

4 + 2) 24.

I

C

(x3−y2)dx+xydy, C l`a cung cu’a du.`o.ng y=ax t`u diˆe’m (0,1) dˆe´n diˆe’m (1, a) (DS

4 + a2

2 +

3(1−a2)

4 lna ) 25.

I

C

x=a(t−sint),y=a(1−cost),a >0 c´o di.nh hu.´o.ng theo hu.´o.ng t˘ang cu’a tham sˆo´ (DS a3π(5−2π))

´

Ap du.ng cˆong th´u.c Green dˆe’ t´ınh t´ıch phˆan du.`o.ng 26.

I

C

xy2dy−x2dx, C l`a du.`o.ng tr`on x2+y2 =a2 (DS πa

4

4 ) 27.

I

C

(x+y)dx−(x−y)dy, C l`a elip x

2

a2 +

y2

b2 = (DS −2πab)

28.

I

C

e−x2+y2(cos 2xydx+ sin 2xydy), C l`a du.`o.ng tr`on x2+y2 =R2.

(DS 0) 29.

I

C

(xy+exsinx+x+y)dx+ (xy−e−y +x−siny)dy,

C l`a du.`o.ng tr`on x2+y2 = 2x. (DS. −π)

30.

I

C

(1 +xy)dx+y2dy,C l`a biˆen cu’a nu.’ a trˆen cu’a h`ınh tr`on x2+y2 62x(y>0). (DS. −π

2) 31.

I

C

(x2+y2)dx+ (x2 −y2)dy, C l`a biˆen cu’a tam gi´ac ∆ABC v´o.i A= (0,0), B = (1,0),C = (0,1), Kiˆe’m tra kˆe´t qua’ b˘a`ng c´ach t´ınh tru c tiˆe´p (DS 0)

32.

I

C

(2xy−x2)dx+ (x+y3)dy, C l`a biˆen cu’a miˆ`n bi ch˘a.n gi´o.i ha.ne bo.’ i hai du.`o.ng y = x2 v`a y2 = x Kiˆe’m tra kˆe´t qua’ b˘a`ng c´ach t´ınh

tru c tiˆe´p (DS 30) 33.

I

C

34.

I

C

(xy+x+y)dx+ (xy+x−y)dy, d´o C l`a a) elip x

2

a2 +

y2

b2 = 1;

b) du.`o.ng tr`on x2+y2=ax (a >0) (DS a) 0; b)−πa

8 ) 35.

I

C

xy2dx−x2ydy,C l`a du.`o.ng tr`on x2+y2 =R2 (DS. πR

2 ) 36.

I

C

2(x2 +y2)dx +x(4y + 3)dy, C l`a du.`o.ng gˆa´p kh´uc v´o.i dı’nh A = (0,0), B = (1,1), C = (0,2) Kiˆe’m tra kˆe´t qua’ b˘a`ng c´ach t´ınh tru c tiˆe´p (DS 3)

Chı’ dˆa˜n. Bˆo’ sung cho C doa.n th˘a’ng dˆe’ thu du.o c chu tuyˆe´n d´ong

37. H˜ay so s´anh hai t´ıch phˆan I1 =

I

AmB

(x+y)2dx−(x−y)2dy v`a I2 =

I

AnB

(x+y)2dx−(x−y)2dy nˆe´uAmBl`a doa.n th˘a’ng nˆo´iA(1,1) v´o.iB(2,6) v`aAnBl`a cung parabol quaA. B v`a gˆo´c to.a dˆo (DS.I1−I2 = 2)

38. T´ınh I =

I

AmBnA

(x+y)dx−(x−y)dy, d´o AmB l`a cung parabol qua A(1,0) v`a B(2,3) v`a c´o tru.c dˆo´i x´u.ng l`a tru.c Oy, c`on AnB l`a doa.n th˘a’ng nˆo´i A v´o.iB

(DS −1

3)

Chı’ dˆa˜n. Dˆ` u tiˆen viˆe´t phu.o.ng tr`ınh parabol v`a du.`o.ng th˘a’ng, saua d´o ´ap du.ng cˆong th´u.c Green

39. Ch´u.ng minh r˘a`ng gi´a tri cu’a t´ıch phˆan

I

C

(2xy−y)dx+x2dy, d´o C l`a chu tuyˆe´n d´ong, b˘a`ng diˆe.n t´ıch miˆe`n ph˘a’ng v´o.i biˆen l`a

40.

I

C

(x+y)2dx−(x2+y2)dy, C l`a biˆen cu’a ∆ABC v´o.i dı’nh A(1,1),B(3,2) v`aC(2,5) (DS −462

3) 41.

I

C

(y−x2)dx+ (x+y2)dy,C l`a biˆen h`ınh qua.t b´an k´ınh R v`a g´ocϕ(0 6ϕ6 π

2) (DS 0) 42.

I

C

y2dx+ (x+y)2dy, C l`a biˆen cu’a h`ınh tam gi´ac ∆ABC v´o.i A(a,0),B(a, a),C(0, a) (DS. 2a

3

3 )

12.4 T´ıch phˆan m˘a.t

12.4.1 C´ac di.nh ngh˜ıa co ba’n

Gia’ su.’ c´ac h`am f(M), P(M), Q(M) v`a R(M),M = (x, y, z) liˆen tu.c ta.i mo.i diˆe’mM cu’a m˘a.t tro.n, du.o c (σ) (m˘a.t tro.n l`a m˘a.t c´o m˘a.t ph˘a’ng tiˆe´p x´uc ta.i mo.i diˆe’m cu’a n´o) Chia mˆo.t c´ach t`uy ´y m˘a.t (σ) th`anh n ma’nh σ0, σ1, , σn−1 v´o.i diˆe.n t´ıch tu.o.ng ´u.ng l`a ∆S0,

∆S1, ,∆Sn−1 D˘a.t dk = diamσk; d = max

06k6n−1dk Trong mˆo˜i ma’nh

m˘a.t ta lˆa´y mˆo.t c´ach t`uy ´y diˆe’m Ni T´ınh gi´a tri cu’a c´ac h`am d˜a cho ta.i diˆe’mNi,i= 0, n−1 Ta k´y hiˆe.u cosα(Ni), cosβ(Ni) v`a cosγ(Ni) l`a c´ac cosin chı’ phu.o.ng cu’a vecto ph´ap tuyˆe´n~n(Ni) ta.i diˆe’mNi cu’a m˘a.t (σ).

X´et hai c´ach lˆa.p tˆo’ng t´ıch phˆan sau.

(I) Lˆa´y gi´a tri. f(Ni) nhˆan v´o.i c´ac phˆ` n tu.a ’ diˆe.n t´ıch m˘a.t ∆S0,

∆S1, ,∆Sn−1 v`a lˆa.p tˆo’ng

n−1

X

i=0

(II) Kh´ac v´o.i c´ach lˆa.p tˆo’ng t´ıch phˆan (I), phu.o.ng ph´ap n`ay ta lˆa´y gi´a tri. P(Ni), Q(Ni) v`a R(Ni) nhˆan khˆong pha’i v´o.i phˆ` na tu.’ diˆe.n t´ıch ∆Si cu’a c´ac ma’nh m˘a.t σi m`a l`a nhˆan v´o.i h`ınh chiˆe´u cu’a c´ac ma’nh d´o lˆen c´ac m˘a.t ph˘a’ng to.a dˆo. Oxy, Oxz v`aOyz, t´u.c l`a lˆa.p c´ac tˆo’ng da.ng

σxy = n−1

X

i=0

P(Ni)m(σxyi ), m(σixy) =proOxy(σi);

σxz = n−1

X

i=0

Q(Ni)m(σxzi ), m(σ i

xz) =proOxz(σi); σyz =

n−1

X

i=0

R(Ni)m(σyzi ), m(σyzi ) =proOyz(σi) D- i.nh ngh˜ıa 12.4.1. Nˆe´u tˆ` n ta.i gi´o.i ha.n h˜u.u ha.no

lim d→0

n−1

X

i=1

f(Ni)∆Si (12.34) khˆong phu thuˆo.c v`ao ph´ep phˆan hoa.ch m˘a.t (σ) th`anh c´ac ma’nh v`a khˆong phu thuˆo.c v`ao c´ach cho.n c´ac diˆe’m trung gian Ni ∈ σi th`ı gi´o.i ha.n d´o go.i l`a t´ıch phˆan m˘a.t theo diˆe.n t´ıch.

K´y hiˆe.u :

ZZ

(σ)

f(x, y, z)dS

D- i.nh ngh˜ıa 12.4.2. C´ac t´ıch phˆan m˘a.t theo to.a dˆo du.o c di.nh ngh˜ıa bo.’ i

ZZ

(σ)

P(M)dxdy def= lim d→0

n−1

X

i=0

P(Ni)m(σxyi ) (12.35)

ZZ

(σ)

Q(M)dxdz def= lim d→0

n−1

X

i=0

Q(Ni)m(σxzi ) (12.36)

ZZ

(σ)

R(M)dydz def= lim d→0

n−1

X

i=0

nˆe´u c´ac gi´o.i ha.n o’ vˆe´ pha’i (12.35)-(12.37) tˆ ` n ta.i h˜u.u ha.n khˆong phu.o thuˆo.c v`ao ph´ep phˆan hoa.ch m˘a.t (σ) v`a c´ach cho.n diˆe’m trung gianNi, i= 0, n−1

T´ıch phˆan m˘a.t theo to.a dˆo da.ng tˆo’ng qu´at

ZZ

(σ)

P(M)dxdy+Q(M)dxdz+R(M)dydz

l`a tˆo’ng cu’a c´ac t´ıch phˆan m˘a.t theo to.a dˆo (12.35), (12.36) v`a (12.37). Nˆe´u (σ) l`a m˘a.t d´ong (k´ın !) th`ı t´ıch phˆan m˘a.ttheo ph´ıa ngo`ai cu’a n´o du.o c k´y hiˆe.u

ZZ

(σ)+

ho˘a.c do.n gia’n l`a

Z Z

(σ)

nˆe´u n´oi r˜o (σ) l`a m˘a.t n`ao;

c`on t´ıch phˆan theo ph´ıa trong du.o..c k´y hiˆe.u

Z Z

(σ)−

ho˘a.c do.n gia’n l`a

ZZ

(σ)

khi d˜a n´oi r˜o (σ) l`a m˘a.t n`ao.

12.4.2 Phu.o.ng ph´ap t´ınh t´ıch phˆan m˘a t

Phu.o.ng ph´ap chung dˆe’ t´ınh t´ıch phˆan m˘a.t ca’ hai da.ng l`a du.a vˆe` t´ıch phˆan hai l´o.p Cu thˆe’ l`a: xuˆa´t ph´at t`u phu.o.ng tr`ınh cu’a m˘a.t (σ) ta biˆe´n dˆo’i biˆe’u th´u.c du.´o.i dˆa´u t´ıch phˆan th`anh biˆe’u th´u.c hai biˆe´n m`a miˆ`n biˆe´n thiˆen cu’a ch´e ung l`a h`ınh chiˆe´u do.n tri cu’a (σ) lˆen m˘a.t ph˘a’ng to.a dˆo tu.o.ng ´u.ng v´o.i c´ac biˆe´n d´o

1+ Nˆe´u m˘a.t (σ) c´o phu.o.ng tr`ınh z = ϕ(x, y) th`ı t´ıch phˆan m˘a.t theo diˆe.n t´ıch du.o c biˆe´n dˆo’i th`anh t´ıch phˆan hai l´o.p theo cˆong th´u.c

dS=

q

1 +ϕ0 x

2

+ϕ0 y

2

dxdy

ZZ

(σ)

f(x, y, z)dS=

ZZ

D(x,y)

f[x, y, ϕ(x, y)]

q

1 +ϕ0 x

2

+ϕ0 y

2

dxdy (12.38)

Nˆe´u m˘a.t (σ) c´o phu.o.ng tr`ınhy =ψ(x, z) th`ı

ZZ

(σ)

f(x, y, z)dS =

Z Z

D(x,z)

f[x, ψ(x, z), z]

q

1 +ψ0 x

2

+ψ0 z

2

dxdz, (12.39)

trong d´o D(x, z) =proOxz(σ)

Nˆe´u m˘a.t (σ) c´o phu.o.ng tr`ınhx =g(y, z) th`ı

ZZ

(σ)

f(·)dS =

ZZ

D(y,z)

f[g(y, z), y, z]

q

1 +g0 y

2+g0 z

2dydz, (12.40)

trong d´o D(y, z) =proOyz(σ) 2+ Gia’ thiˆe´t m˘

a.t (σ) chiˆe´u du.o c do.n tri lˆen c´ac m˘a.t ph˘a’ng to.a dˆo., t´u.c l`a m˘a.t c´o phu.o.ng tr`ınh da.ng

z =ϕ(x, y), (x, y)∈D(x, y); y=ψ(x, z), (x, z)∈D(x, z); x=g(y, z), (y, z)∈D(y, z).

Ta k´y hiˆe.ue1,e2,e3l`a c´ac vecto co so.’ cu’aR3v`a cosα(M) = cos(~n, ~de1),

cosβ(M) = cos(~n, ~de2), cosγ(M) = cos(~n, ~de3) D´o l`a c´ac cosin chı’

phu.o.ng cu’a vecto ph´ap tuyˆe´n v´o.i m˘a.t (σ) ta.i diˆe’mM ∈(σ) Khi d´o c´ac t´ıch phˆan m˘a.t theo to.a dˆo lˆa´y theo m˘a.t hai ph´ıa du.o c t´ınh nhu. sau

ZZ

(σ)

P(M)dxdy =

            

+

ZZ

D(x,y)

P(x, y, ϕ(x, y))dxdy nˆe´u cosγ >0;

−

ZZ

D(x,y)

P(x, y, ϕ(x, y))dxdy nˆe´u cosγ <0

Tu.o.ng tu ta c´o

ZZ

(σ)

Q(M)dxdz =

            

+

ZZ

D(x,z)

Q(x, ψ(x, z), z)dxdz nˆe´u cosβ >0,

−

ZZ

D(x,z)

Q(·)dxdz nˆe´u cosβ <0;

Z Z

(σ)

R(M)dydz =

            

+

ZZ

D(y,z)

R(g(y, z), y, z)dydz nˆe´u cosα >0

−

ZZ

D(y,z)

R(·)dydz nˆe´u cosα <0

Nhˆa n x´et. T´ıch phˆan m˘a.t theo to.a dˆo lˆa´y theo phˆa` nm˘a t tru. v´o.i du.`o.ng sinh song song v´o.i tru.c Oz l`a b˘a`ng Trong c´ac tru.`o.ng ho p tu.o.ng tu , c´ac t´ıch phˆan m˘a.t theo to.a dˆo x,z hay y,z c˜ung =

12.4.3 Cˆong th´u.c Gauss-Ostrogradski D´o l`a cˆong th´u.c

ZZZ

D

∂P

∂x + ∂Q

∂y + ∂R

∂z

dxdydz =

Z Z

∂D

P dydz+Qdxdz+Rdxdy.

N´o x´ac lˆa.p mˆo´i liˆen hˆe gi˜u.a t´ıch phˆan m˘a.t theo m˘a.t biˆen ∂D cu’a D v´o.i t´ıch phˆan 3-l´o.p lˆa´y theo miˆ`ne D ⊂R3

12.4.4 Cˆong th´u.c Stokes D´o l`a cˆong th´u.c

I

L

P dx+Qdy+Rdz=

ZZ

(σ)

∂Q

∂x − ∂P

∂y

dxdy+∂R ∂y −

∂Q ∂z

dydz

+

∂P

∂z − ∂R

∂x

N´o x´ac lˆa.p mˆo´i liˆen hˆe gi˜u.a t´ıch phˆan m˘a.t theo m˘a.t (σ) v´o.i t´ıch phˆan du.`o.ng lˆa´y theo b`o.L cu’a m˘a.t (σ).

Ta nhˆa.n x´et r˘a`ng sˆo´ ha.ng th´u nhˆa´t o.’ vˆe´ pha’i cu’a cˆong th´u.c Stokes c˜ung ch´ınh l`a vˆe´ pha’i cˆong th´u.c Green Hai sˆo´ ha.ng c`on la.i thu du.o c t`u d´o bo.’ i ph´ep ho´an vi tuˆa` n ho`an c´ac biˆe´nx, y, z v`a c´ac h`am P, Q, R:

x

% &

z ←− y

P

% &

R ←− Q C ´AC V´I DU.

V´ı du 1. T´ınh t´ıch phˆan

ZZ

(σ)

(6x+ 4y+ 3z)dS, d´o (σ) l`a phˆ` na m˘a.t ph˘a’ng x+ 2y+ 3z = n˘a`m g´oc phˆa` n t´am th´u nhˆa´t

Gia’i. M˘a.t t´ıch phˆan l`a tam gi´ac ABC tronng d´o A(6,0,0), B(0,3,0) v`a C(0,0,2) Su.’ du.ng phu.o.ng tr`ınh cu’a (σ) dˆe’ biˆe´n dˆo’i t´ıch phˆan m˘a.t th`anh t´ıch phˆan 2-l´o.p T`u phu.o.ng tr`ınh cu’a (σ) r´ut z =

3(6−x−2y) T`u d´o dS =

q

1 +z0 x

2

+z0 y

2

dxdy=

√

14 dxdy. Do d´o

I =

√

14

ZZ

∆OAB

[(6x+ 4y+3

3(6−x−2y)]dxdy =

√

14

3

Z

0

dy

6Z−2y

0

(5x+ 2y+ 6)dx

=

√

14

3

Z

0

nh5

2x

2

+ 2xy+ 6xi

6−2y

0

o

dy= 54

√

V´ı du 2. T´ınh

ZZ

(σ)

p

1 + 4x2+ 4y2dS, (σ) l`a phˆ` n paraboloid tr`ona

xoayz = 1−x2 −y2 n˘a`m trˆen m˘a.t ph˘a’ng Oxy.

Gia’i. M˘a.t (σ) chiˆe´u du.o c do.n tri lˆen m˘a.t ph˘a’ngOxy v`a h`ınh tr`on x2+y2 61 l`a h`ınh chiˆe´u cu’a n´o: D(x, y) =(x, y) :x2+y2 61 Ta

t´ınh dS Ta c´o zx0 = −2x, z0y = −2y ⇒ dS = p1 + 4x2+ 4y2dxdy.

Do vˆa.y

ZZ

(σ)

=

Z Z

D(x,y)

p

1 + 4x2+ 4y2 ·p1 + 4x2 + 4y2dxdy

=

ZZ

x2+y261

(1 + 4x2+ 4y2)dxdy

B˘a`ng c´ach chuyˆe’n sang to.a dˆo cu c ta c´o

I =

2π

Z

0

dϕ

1

Z

0

(1 + 4r2)rdr = 3π N

V´ı du 3. T´ınh t´ıch phˆan

ZZ

(σ)

(y2+z2dxdy, d´o (σ) l`a ph´ıa ngo`ai cu’a m˘a.t z =

√

1−x2 gi´o.i ha.n bo’ i c´ ac m˘a.t ph˘a’ng y= 0, y= 1.

Gia’i. M˘a.t (σ) l`a nu.’ a trˆen cu’a m˘a.t tru. x2+z2 = 1, z > Do d´o h`ınh chiˆe´u cu’a (σ) lˆen m˘a.t ph˘a’ng Oxy l`a h`ınh ch˜u nhˆa.t x´ac di.nh bo.’i c´ac diˆ`u kiˆe.n:e −1 x 1, y Do d´o v`ız =

√

1−x2 nˆen

cosγ >0 v`a

Z Z

(σ)

(y2+z2)dxdy =

ZZ

D(x,y)

[y2+ (

√

1−x2)2

]dxdy

=

1

Z

−1

dx

1

Z

0

V´ı du 4. T´ınh t´ıch phˆan

ZZ

(σ)

2dxdy+ydxdz−x2zdydz, d´o (σ) l`a ph´ıa trˆen cu’a phˆ` n elipxoid 4xa 2+y2+ 4z2 = n˘a`m g´oc phˆa` n

t´am I

Gia’i. Ta viˆe´t t´ıch phˆan d˜a cho du.´o.i da.ng

I =

Z Z

(σ)

dxdy+

ZZ

(σ)

ydydz−

ZZ

(σ)

x2zdydz.

v`a su.’ du.ng phu.o.ng tr`ınh cu’a m˘a.t (σ) dˆe’ biˆe´n dˆo’i mˆo˜i t´ıch phˆan Lu.u ´

y r˘a`ng cosα >0, cosβ >0, cosγ >0

(i) V`ı h`ınh chiˆe´u cu’a m˘a.t (σ) lˆen m˘a.t ph˘a’ng Oxy l`a phˆ` n tu h`ınha elip x

2

12 +

y2

22 61 nˆen

I1 =

ZZ

(σ)

dxdy=

ZZ

D(x,y)

dxdy= π

2 (v`ı diˆe.n t´ıch elip = 2π)

(ii) H`ınh chiˆe´u cu’a (σ) lˆen m˘a.t ph˘a’ng Oxz l`a phˆ` n tu h`ınh tr`ona 4x2+ 4z2 64⇔ x2+z2 61 M˘a.t kh´ac t`u phu.o.ng tr`ınh m˘a.t r´ut ra y= 2p1−x2−y2 v`a d´o

I2 =

Z Z

(σ)

ydxdz=

Z Z

D(x,y)

√

1−x2−z2dxdz =|chuyˆe’n sang to.a dˆo cu c|

= π/2

Z

0

dϕ

1

Z

0

√

1−r2rdr = π

3·

(iii) H`ınh chiˆe´u cu’a (σ) lˆen m˘a.t ph˘a’ng Oyz l`a mˆo.t phˆa` n tu h`ınh elip y

2

4 +z

2

x=

r

1− y

2

4 −z

2 rˆ` i thˆe´ v`ao h`am du.´o.i dˆa´u t´ıch phˆan cu’ao I 3:

I3 =

ZZ

(σ)

x2zdydz =

Z Z

D(y,z)

z 1− y

2

4 −z

2

dydz

=

1

Z

0

dz

2√1−z2

Z

0

z 1− y

2

4 −z

2

dy =· · ·= 15 · Nhu vˆa.y I = 2I1+I2−I3 =

4π −

4 15· N V´ı du 5. T´ınh

ZZ

(σ)−

ydydz, d´o (σ) l`a m˘a.t cu’a t´u diˆe.n gi´o.i ha.n bo.’ i m˘a.t ph˘a’ng x+y+z = v`a c´ac m˘a.t ph˘a’ng to.a dˆo., t´ıch phˆan du.o c lˆa´y theo ph´ıa cu’a t´u diˆe.n.

Gia’i. M˘a.t ph˘a’ng x+y+z = c˘a´t c´ac tru.c to.a dˆo ta.iA(1,0,0), B(0,1,0) v`a C = (0,0,1) Ta k´y hiˆe.u gˆo´c to.a dˆo l`a O(0,0,0) T`u d´o suy m˘a.t k´ın (σ) gˆ` m t`o u h`ınh tam gi´ac ∆ABC, ∆BCO, ∆ACO v`a ∆ABO Do vˆa.y t´ıch phˆan d˜a cho l`a tˆo’ng cu’a bˆo´n t´ıch phˆan.

(i) T´ıch phˆan I1 =

ZZ

ABC

ydxdz R´ut y t`u phu.o.ng tr`ınh m˘a.t (σ)⊃

∆ABC ta c´oy= 1−x−z v`a d´o I1 =−

Z Z

ACO

(1−x−z)dxdz =

1

Z

0

dx

1−x

Z

0

(x+z−1)dz =−1

6· (Lu.u ´y r˘a`ng cosβ = cos(~n, Oy)< v`ı vecto.~n lˆa.p v´o.i hu.´o.ng du.o.ng tru.cOy mˆo.t g´oc t`u, d´o tru.´o.c t´ıch phˆan theo ∆ACOxuˆa´t hiˆe.n dˆa´u tr`u.)

(ii)

ZZ

(BCD)

ydxdz=

ZZ

(ABO)

v`ı m˘a.t ph˘a’ng BCO v`aABO dˆ`u vuˆong g´oc v´o.i m˘a.t ph˘a’nge Oxz. (iii)

ZZ

(ACO)

ydxdz =

Z Z

ACO

0dxdz =

Vˆa.y I =−1

6 N

V´ı du 6. T´ınh t´ıch phˆan I =

ZZ

(σ)

x3dydz+y3dzdx+z3dxdy, trong d´o (σ) l`a ph´ıa ngo`ai m˘a.t cˆa` ux2 +y2+z2 =R2.

Gia’i. Ap du.ng cˆong th´u.c Gauss-Ostrogradski ta c´o´

ZZ

(σ)

=

Z ZZ

D

(x2+y2+z2)dxdydz

trong d´o D ⊂ R3 l`a miˆ`n v´o.i biˆen l`a m˘a.t (e σ) Chuyˆe’n sang to.a dˆo. cˆ` u ta c´oa

3

ZZ Z

D

(x2 +y2+z2)dxdydz =

2π

Z

0

dϕ π

Z

0

sinθdθ R

Z

0

r4dr

= 12πR

5

5 · Vˆa.y I = 12πR

5

5 · N V´ı du 7. T´ınh t´ıch phˆan

I

L

x2y3dx+dy+zdz, d´o L l`a du.`o.ng tr`on x2+y2 = 1, z = 0, c`on m˘

a.t (σ) l`a ph´ıa ngo`ai cu’a nu.’ a m˘a.t cˆa` u x2+y2+z2 = 1, z >0 v`aL c´

o di.nh hu.´o.ng du.o.ng

Gia’i. Trong tru.`o.ng ho p n`ay P =x2y3, Q= 1, R =z Do d´o ∂Q

∂x − ∂P

∂y =−3x

2

y2, ∂R ∂y −

∂Q ∂z = 0,

∂P ∂z −

v`a d´o theo cˆong th´u.c Stokes ta c´o

I

L

=−3

Z Z

(σ)

x2y2dxdy=−π

8· N B `AI T ˆA P

T´ınh c´ac t´ıch phˆan m˘a.t theo diˆe.n t´ıch sau dˆay 1.

ZZ

(Σ)

(x+y+z)dS, (Σ) l`a m˘a.t lˆa.p phu.o.ng 0 6x61, 0661, 06z 61 (DS 9)

2.

ZZ

(Σ)

(2x+y+z)dS, (Σ) l`a phˆ` n m˘a.t ph˘a’nga x+y+z = n˘a`m

g´oc phˆ` n t´ama I (DS.

√

3 ) 3.

ZZ

(Σ)

z + 2x+ 4y

dS, (Σ) l`a phˆ` n m˘a.t ph˘a’ng 6a x+ 4y+ 3z = 12 n˘a`m g´oc phˆa` n t´am I (DS 4√61)

4.

ZZ

(σ)

p

x2+y2dS, (Σ) l`a phˆ` n m˘a.t n´ona z2 =x2+y2, 0 6z 61.

(DS

√

2π ) 5.

ZZ

(Σ)

(y+z+

√

a2 −x2)dS, (Σ) l`a phˆ` n m˘a.t tru.a x2

+y2 =a2 n˘a`m gi˜u.a hai m˘a.t ph˘a’ng z = v`a z =h (DS.ah(4a+πh))

6.

ZZ

(Σ)

p

y2−x2dS, (Σ) l`a phˆ` n m˘a.t n´ona z2=x2+y2 n˘a`m m˘a.t

tru x2+y2 =a2 (DS 8a

3

7.

ZZ

(Σ)

(x+y+z)dS, (Σ) l`a nu.’ a trˆen cu’a m˘a.t cˆa` u x2+y2+z2 =a2 (DS πa3)

8.

ZZ

(Σ)

p

x2+y2dS, (Σ) l`a m˘a.t cˆa` u x2+y2+z2 =a2 (DS. 8πa

3 ) 9.

ZZ

(Σ)

dS

(1 +x+y), (Σ) l`a biˆen cu’a t´u diˆe.n x´ac di.nh bo

’ i bˆa´t phu.o.ng

tr`ınhx+y+z 61,x>0,y>0,z>0 (DS 3(3−

√

3)+(√3−1) ln 2) 10.

Z Z

(Σ)

(x2 +y2)dS, (Σ) l`a phˆ` n m˘a.t paraboloida x2+y2 = 2z

du.o c

c˘a´t bo.’ i m˘a.t ph˘a’ng z= (DS 55 +

√

3 65 ) 11.

Z Z

(Σ)

p

1 + 4x2+ 4y2dS, (Σ) l`a phˆ` n m˘a.t paraboloida z = 1−x2−y2

gi´o.i ha.n bo.’i c´ac m˘a.t ph˘a’ng z = v`a z= (DS 3π) 12.

Z Z

(Σ)

(x2 +y2)dS, (Σ) l`a phˆ` n m˘a.t n´ona z = px2+y2 n˘a`m gi˜u.a

c´ac m˘a.t ph˘a’ngz = v`a z= (DS π

√

2 ) 13.

Z Z

(Σ)

(xy+yz+zx)dS, (Σ) l`a phˆ` n m˘a.t n´ona z = px2+y2 n˘a`m

trong m˘a.t tru. x2+y2 = 2ax (a >0) (DS 64a

4√2

15 ) 14.

ZZ

(Σ)

(x2+y2+z2)dS, (Σ) l`a ma.t cˆa` u (DS 4π)

15.

ZZ

(Σ)

bo.’ i m˘a.t ph˘a’ng x= 10 (DS 50π

3 (1 + 25

√

5)) Su.’ du.ng cˆong th´u.c t´ınh diˆe.n t´ıch m˘a.tS(Σ) =

Z Z

(Σ)

dS dˆe’ t´ınh diˆe.n t´ıch cu’a phˆ` n m˘a.t (Σ) nˆe´ua

16. (Σ) l`a phˆ` n m˘a.t ph˘a’ng 2a x+ 2y+z = 8a n˘a`m m˘a.t tru. x2+y2 =R2. (DS 3πR2)

17. (Σ) l`a phˆ` n m˘a.t tru.a y+z2 =R2 n˘a`m m˘a.t tru.

x2+y2 =R2. (DS 8R2)

18. (Σ) l`a phˆ` n m˘a.t paraboloida x2+y2 = 6z n˘a`m m˘a.t tru x2+y2 = 27. (DS 42π)

19. (Σ) l`a phˆ` n m˘a.t cˆaa ` u x2+y2+z2 = 3a2 n˘a`m paraboloid x2+y2 = 2az. (DS 2πa2(3−√3))

20. (Σ) l`a phˆ` n m˘a.t n´ona z2 = 2xy n˘a`m g´oc phˆa` n t´am I gi˜u.a hai m˘a.t ph˘a’ng x= 2, y= (DS 16)

21. (Σ) l`a phˆ` n m˘a.t tru.a x2+y2 =Rx n˘a`m m˘a.t cˆa` u

x2+y2+z2 =R2. (DS 4R2)

T´ınh c´ac t´ıch phˆan m˘a.t theo to.a dˆo sau: 22.

ZZ

(Σ)

dxdy, (Σ) l`a ph´ıa ngo`ai phˆ` n m˘a.t n´ona z =px2+y2 khi

06z 61 (DS −π) 23.

ZZ

(Σ)

ydzdx, (Σ) l`a ph´ıa trˆen cu’a phˆ` n m˘a.t ph˘a’nga x+y+z = a

(a >0) n˘a`m g´oc phˆa` n t´am I (DS. a

3

6) 24.

ZZ

(Σ)

25.

ZZ

(Σ)

−xdydz+zdzdx + 5dxdy, (Σ) l`a ph´ıa trˆen cu’a phˆ` n m˘a.ta ph˘a’ng 2x+ 3y+z = thuˆo.c g´oc phˆa` n t´am I (DS 6)

26.

Z Z

(Σ)

yzdydz+xzdxdz+xydxdy, (Σ) l`a ph´ıa trˆen cu’a tam gi´ac ta.o bo.’ i giao tuyˆe´n cu’a m˘a.t ph˘a’ng x+y+z = a v´o.i c´ac m˘a.t ph˘a’ng to.a dˆo (DS. a

4

8 ) 27.

ZZ

(Σ)

x2dydz+z2dxdy, (Σ) l`a ph´ıa ngo`ai cu’a phˆ` n m˘a.t n´ona

x2+y2 =z2, 06z 61. (DS. −4

3) 28.

ZZ

(Σ)

xdydz+ydzdx+zdxdy, (Σ) l`a ph´ıa ngo`ai phˆ` n m˘a.t cˆaa ` u x2+y2+z2=a2 (DS 4πa3)

29.

ZZ

(σ)

x2dydz−y2dzdx+z2dxdy, (Σ) l`a ph´ıa ngo`ai cu’a m˘a.t cˆa` u x2+y2+z2=R2 thuˆo.c g´oc phˆa` n t´am I (DS πa

4

8 ) 30.

ZZ

(Σ)

2dxdy+ydzdx−x2zdydz, (Σ) l`a ph´ıa ngo`ai cu’a phˆ` n m˘a.ta elipxoid 4x2 +y2+ 4z2 = thuˆ

o.c g´oc phˆa` n t´am I (DS 4π −

4 15) 31.

ZZ

(Σ)

(y2+z2)dxdy, (Σ) l`a ph´ıa ngo`ai cu’a m˘a.t tru. z2 = 1−x2,

06y61 (DS π 3) 32.

ZZ

(Σ)

(z−R)2dxdy, (Σ) l`a ph´ıa ngo`ai cu’a nu.’ a m˘a.t cˆa` u x2+y2+ (z−R)2 =R2,R 6z 62R (DS −5π

33.

ZZ

(Σ)

x2dydz+y2dzdx+z2dxdy, (Σ) l`a ph´ıa ngo`ai cu’a phˆ` n m˘a.ta

cˆ` ua x2+y2+z2 =a2 thuˆo.c g´oc phˆa` n t´am I (DS 3πa

4

8 ) 34.

ZZ

(Σ)

z2dxdy, (σ) l`a ph´ıa cu’a m˘a.t elipxoid x2+y2+ 2z2 = (DS 0)

35.

ZZ

(Σ)

(z+ 1)dxdy, (Σ) l`a ph´ıa ngo`ai cu’a m˘a.t cˆa` u

x2+y2+z2 =R2 (DS. 4πR

3 ) 36.

ZZ

(Σ)

x2dydz+y2dzdx+z2dxdy, (Σ) l`a ph´ıa ngo`ai cu’a m˘a.t cˆa` u

(x−a)2+ (y−b)2+ (z−c)2 =R2 (DS. 8πR

3 (a+b+c)) 37.

ZZ

(Σ)

x2y2zdxdy, (Σ) l`a ph´ıa cu’a nu.’ a du.´o.i m˘a.t cˆa` u

x2+y2+z2 =R2 (DS. 2πR

105 ) 38.

ZZ

(Σ)

xzdxdy+xydydz+yzdxdz, (Σ) l`a ph´ıa ngo`ai cu’a t´u diˆe.n ta.o

bo.’ i c´ac m˘a.t ph˘a’ng to.a dˆo v`a m˘a.t ph˘a’ng x+y+z = (DS 8)

Chı’ dˆa˜n. Su.’ du.ng nhˆa.n x´et nˆeu phˆa` n l´y thuyˆe´t 39.

ZZ

(Σ)

yzdydz+xzdxdz +xydxdy, (Σ) l`a ph´ıa ngo`ai cu’a m˘a.t biˆen t´u diˆe.n lˆa.p bo.’i c´ac m˘a.t ph˘a’ng x = 0, y = 0, z = 0, x+y+z = a. (DS 0)

40.

ZZ

(Σ)

m˘a.t cˆa` u x2+y2 +z2 =R2 (z >0) (DS πR

4

2 ) ´

Ap du.ng cˆong th´u.c Gauss-Ostrogradski dˆe’ t´ınh t´ıch phˆan m˘a.t theo ph´ıa ngo`ai cu’a m˘a.t (Σ) (nˆe´u m˘a.t khˆong k´ın th`ı bˆo’ sung dˆe’ n´o tro.’ th`anh k´ın)

41.

ZZ

(Σ)

x2dydz+y2dzdx+z2dxdy, (Σ) l`a m˘a.t cˆa` u

(x−a)2+ (y−b)2+ (z−c)2 =R2 (DS 8π

3 (a+b+c)R

3

) 42.

ZZ

(Σ)

xdydz+ydzdx+zdxdy, (Σ) l`a m˘a.t cˆa` u x2+y2+z2 =R2.

(DS 4πR3)

43.

ZZ

(Σ)

4x3dydz+ 4y3dzdx−6z2dxdy, (Σ) l`a biˆen cu’a phˆ` n h`ınha tru x2+y2 6a2, 0 6z 6h. (DS 6πa2(a2−h2))

44.

ZZ

(σ)

(y−z)dydz+ (z−x)dzdx+ (x−y)dxdy, (Σ) l`a phˆ` n m˘a.ta n´on x2+y2 =z2, 06x6h (DS 0)

Chı’ dˆa˜n. V`ı (Σ) khˆong k´ın nˆen cˆ` n bˆo’ sung phˆaa ` n m˘a.t ph˘a’ngz =h n˘a`m n´on dˆe’ thu du.o c m˘a.t k´ın.

45.

ZZ

(Σ)

dydz+zxdzdx+xydxdy, (Σ) l`a biˆen cu’a miˆ`ne

{(x, y, z) :x2+y2 6a2,06z 6h} (DS 0)

46.

ZZ

(Σ)

ydydz+zdzdx+xdxdy, (Σ) l`a m˘a.t cu’a h`ınh ch´op gi´o.i ha.n bo.’ i c´ac m˘a.t ph˘a’ng

x+y+z =a (a >0),x= 0, y= 0, z = (DS 0) 47.

ZZ

(Σ)

(DS π 5) 48.

Z Z

(Σ)

x3dydz+y3dzdx+z3dxdy, (Σ) l`a m˘a.t cˆa` ux2+y2+z2 =a2.

(DS 12πa

5

5 ) 49.

ZZ

(Σ)

z2dxdy, (Σ) l`a m˘a.t elipxoid x

2

a2 +

y2

b2 +

z2

c2 = (DS 0)

Chı’ dˆa˜n. Xem v´ı du 10, mu.c III.

50.

ZZ

(Σ)

xdydz+ydzdx+zdxdy, (Σ) l`a m˘a.t elipxoidx

2

a2+

y2

b2 +

z2

c2 =

(Ds 4πabc) 51.

ZZ

(Σ)

xdydz+ydzdx+zdxdy, (Σ) l`a biˆen h`ınh tru.x2+y2 6a2,

−h6z 6h. (DS 6πa2h) 52.

ZZ

(Σ)

x2dydz+y2dzdx+z2dxdy, (Σ) l`a biˆen cu’a h`ınh lˆa.p phu.o.ng 06x6a, 06y6a, 06z 6a. (DS 3a4)

Dˆe’ ´ap du.ng cˆong th´u.c Stokes, ta lu.u ´y la.i quy u.´o.c

Hu.´o.ng du.o.ng cu’a chu tuyˆe´n∂Σ cu’a m˘a.t (Σ) du.o c quy u.´o.c nhu sau: Nˆe´u mˆo.t ngu.`o.i quan tr˘a´c d´u.ng trˆen ph´ıa du.o c cho.n cu’a m˘a.t (t´u.c l`a hu.´o.ng t`u chˆan dˆe´n dˆ` u tr`a ung v´o.i hu.´o.ng cu’a vecto ph´ap tuyˆe´n) th`ı ngu.`o.i quan s´at di chuyˆe’n trˆen∂Σ theo hu.´o.ng d´o th`ı m˘a.t (Σ) luˆon luˆon n˘a`m bˆen tr´ai

´

Ap du.ng cˆong th´u.c Stokes dˆe’ t´ınh c´ac t´ıch phˆan sau 53.

I

C

54.

I

C

ydx+zdy+xdz,C l`a du.`o.ng tr`on x2+y2+z2 =R2,x+y+z = c´o hu.´o.ng ngu.o..c chiˆe`u kim dˆo`ng hˆo` nˆe´u nh`ın t`u phˆa` n du.o.ng tru.c Ox. (DS −√3πR2)

55.

I

C

(y− z)dx+ (z − x)dy + (x− y)dz, C l`a elip x2 +y2 = a2, x

a + z

h = (a > 0, h > 0) c´o hu.´o.ng ngu.o c chiˆe`u kim dˆo` ng hˆo` nˆe´u nh`ın t`u diˆe’m (2a,0,0) (DS −2πa(a+h))

56.

I

C

(y−z)dx+(z−x)dy+(x−y)dz,C l`a du.`o.ng tr`onx2+y2+z2 =a2,

y=xtgα, 0 < α < π

2 c´o hu.´o.ng ngu.o c chiˆe`u kim dˆo` ng hˆo` nh`ın t`u diˆe’m (2a,0,0) (DS 2√2πa2sinπ

4 −α)) 57.

I

C

(y−z)dx+ (z−x)dy+ (x−y)dz,C l`a elipx2+y2 = 1,x+z = c´o hu.´o.ng ngu.o c chiˆe`u kim dˆo` ng hˆo` nˆe´u nh`ın t`u phˆ` n du.o.ng tru.ca Oz. (DS −4π)

58.

I

C

(y2−z2)dx+ (z2−x2)dy+ (x2−y2)dz,C l`a biˆen cu’a thiˆe´t diˆe.n cu’a lˆa.p phu.o.ng 0 x a, 0 y a, 0 z a v´o.i m˘a.t ph˘a’ng x+y+z = 3a

2 c´o hu.´o.ng ngu.o c chiˆe`u kim dˆo` ng hˆo` nˆe´u nh`ın t`u diˆe’m (2a,0,0) (DS −9

2a

3

) 59.

I

C

exdx+z(x2 +y2)3/2dy+yz3dz, C l`a giao tuyˆe´n cu’a m˘a.t z =

p

x2+y2 v´o.i c´ac m˘

a.t ph˘a’ng x= 0, x= 2, y = 0, y= (DS −14)

60.

I

C

8yp(1−x2−z2)3dx+xy3

L´y thuyˆe´t chuˆo˜i

13.1 Chuˆo˜i sˆo´ du.o.ng 178

13.1.1 C´ac di.nh ngh˜ıa co ba’n 178 13.1.2 Chuˆo˜i sˆo´ du.o.ng 179

13.2 Chuˆo˜i hˆo i tu tuyˆe.t dˆo´i v`a hˆo.i tu khˆong tuyˆe.t dˆo´i 191

13.2.1 C´ac di.nh ngh˜ıa co ba’n 191 13.2.2 Chuˆo˜i dan dˆa´u v`a dˆa´u hiˆe.u Leibnitz 192

13.3 Chuˆo˜i l˜uy th`u.a 199

13.3.1 C´ac di.nh ngh˜ıa co ba’n 199 13.3.2 D- iˆe`u kiˆe.n khai triˆe’n v`a phu.o.ng ph´ap khai

triˆe’n 201

13.4 Chuˆo˜i Fourier 211

13.1 Chuˆo˜i sˆo´ du.o.ng 13.1.1 C´ac di.nh ngh˜ıa co ba’n Gia’ su.’ cho d˜ay sˆo´ (an) Biˆe’u th´u.c da.ng

a1+a2+· · ·+an+· · ·=

∞

X

n=1

an=X n>1

an (13.1)

du.o c go.i l`a chuˆo˜i sˆo´(hay do.n gia’n l`a chuˆo˜i) C´ac sˆo´a1, , an,

du.o c go.i l`ac´ac sˆo´ ha ng cu’a chuˆo˜i, sˆo´ ha.ngango.i l`a sˆo´ ha ng tˆo’ng qu´at

cu’a chuˆo˜i Tˆo’ng n sˆo´ ha.ng dˆa` u tiˆen cu’a chuˆo˜i du.o c go.i l`a tˆo’ng riˆeng th´u.n cu’a chuˆo˜i v`a k´y hiˆe.u l`a sn, t´u.c l`a

sn=a1+a2+· · ·+an.

V`ı sˆo´ sˆo´ ha.ng cu’a chuˆo˜i l`a vˆo ha.n nˆen c´ac tˆo’ng riˆeng cu’a chuˆo˜i lˆa.p th`anh d˜ay vˆo ha.n c´ac tˆo’ng riˆeng s1, s2, , sn,

D- i.nh ngh˜ıa 13.1.1. Chuˆo˜i (13.1) du.o c go.i l`a chuˆo˜i hˆo.i tu. nˆe´u d˜ay c´ac tˆo’ng riˆeng (sn) cu’a n´o c´o gi´o.i ha n h˜u.u ha n v`a gi´o.i ha.n d´o du.o c go.i l`atˆo’ng cu’a chuˆo˜i hˆo.i tu Nˆe´u d˜ay (sn) khˆong c´o gi´o.i ha.n h˜u.u ha.n th`ı chuˆo˜i (13.1)phˆan k`y

D- i.nh l´y 13.1.1. Diˆ`u kiˆe.n cˆae ` n dˆe’ chuˆo˜i (13.1) hˆo i tu l`a sˆo´ ha ng tˆo’ng qu´at cu’a n´o dˆ` n dˆe´n khia n→ ∞, t´u.c l`a lim

n→∞an= 0.

Di.nh l´y 13.1.1 chı’ l`a diˆ`u kiˆe.n cˆae ` n ch´u khˆong l`a diˆ`u kiˆe.n du’.e Nhu.ng t`u d´o c´o thˆe’ r´ut diˆ`u kiˆe.n du’ dˆe’ chuˆo˜i phˆan k`y:e Nˆe´u

lim n→∞an

6

= th`ı chuˆo˜i P

n>1

an phˆan k`y.

Chuˆo˜i P n>m+1

anthu du.o c t`u chuˆo˜i P n>1

an sau c˘a´t bo’ m sˆo´ ha.ng dˆ` u tiˆen du.o c go.i l`aa phˆ` n du th´a u.mcu’a chuˆo˜i P

n>1

(13.1) hˆo.i tu v`a tˆo’ng cu’a n´o b˘a`ngRm th`ıs=sm+Rm Chuˆo˜i hˆo.i tu c´o c´ac t´ınh chˆa´t quan tro.ng l`a

(i) V´o.i sˆo´m cˆo´ di.nh bˆa´t k`y chuˆo˜i (13.1) v`a chuˆo˜i phˆa` n du th´u.m cu’a n´o dˆ` ng th`o.i hˆo.i tu ho˘a.c dˆoo ` ng th`o.i phˆan k`y

(ii) Nˆe´u chuˆo˜i (13.1) hˆo.i tu th`ıRm →0 m→ ∞

(iii) Nˆe´u c´ac chuˆo˜i P n>1

an v`a P n>1

bn hˆo.i tu v`a α, β l`a h˘a`ng sˆo´ th`ı

X

n>1

(αan+βbn) =αX n>1

an+βX n>1

bn.

13.1.2 Chuˆo˜i sˆo´ du.o.ng Chuˆo˜i sˆo´ P

n>1

an du.o c go.i l`a chuˆo˜i sˆo´ du.o.ng nˆe´u an >0 ∀n∈N Nˆe´u an>0∀n th`ı chuˆo˜i du.o c go.i l`a chuˆo˜i sˆo´ du.o.ng thu c su

Tiˆeu chuˆa’n hˆo i tu . Chuˆo˜i sˆo´ du.o.ng hˆo.i tu v`a chı’ d˜ay tˆo’ng riˆeng cu’a n´o bi ch˘a.n trˆen.

Nh`o diˆ`u kiˆe.n n`ay, ta c´o thˆe’ thu du.o c nh˜u.ng dˆa´u hiˆe.u du’ sau dˆay:e

Dˆa´u hiˆe.u so s´anh I Gia’ su.’ cho hai chuˆo˜i sˆo´ A:X

n>1

an, an>0∀n ∈N v`a B :X n>1

bn, bn>0 ∀n∈N v`aan6bn ∀n∈N Khi d´o:

(i) Nˆe´u chuˆo˜i sˆo´B hˆo.i tu th`ı chuˆo˜i sˆo´A hˆo.i tu., (ii) Nˆe´u chuˆo˜i sˆo´A phˆan k`y th`ı chuˆo˜i sˆo´B phˆan k`y

Dˆa´u hiˆe.u so s´anh II. Gia’ su.’ c´ac chuˆo˜i sˆo´A v`a B l`a nh˜u.ng chuˆo˜i sˆo´ du.o.ng thu..c su v`a ∃ lim

n→∞ an

bn = λ (r˜o r`ang l`a λ +∞) Khi d´o:

(i) Nˆe´u λ <∞th`ı t`u su hˆo.i tu cu’a chuˆo˜i sˆo´B k´eo theo su hˆo.i tu. cu’a chuˆo˜i sˆo´A

(iii) Nˆe´u 0< λ <+∞th`ı hai chuˆo˜i A v`a B dˆ` ng th`o.i hˆo.i tu ho˘a.co dˆ` ng th`o.i phˆan k`o y

Trong thu c h`anh dˆa´u hiˆe.u so s´anh thu.`o.ng du.o c su.’ du.ng du.´o.i da.ng “ thu c h`anh” sau dˆay:

Dˆa´u hiˆe.u thu c h`anh. Nˆe´u dˆo´i v´o.i d˜ay sˆo´ du.o.ng (an) tˆ` n ta.i c´ac sˆo´o pv`aC >0 choan∼ C

np,n → ∞th`ı chuˆo˜i

P

n>1

an hˆo.i tu nˆe´u p >1 v`a phˆan k`y nˆe´u p61

C´ac chuˆo˜i thu.`o.ng du.o c d`ung dˆe’ so s´anh l`a 1) Chuˆo˜i cˆa´p sˆo´ nhˆan P

n>0

aqn, a6= hˆ

o.i tu 06 q <1 v`a phˆan k`y q>1

2) Chuˆo˜i Dirichlet: P n>1

1

nα hˆo.i tu khiα >1 v`a phˆan k`y khiα6 Chuˆo˜i phˆan k`y P

n>1

1

n go.i l`a chuˆo˜i diˆe`u h`oa

T`u dˆa´u hiˆe.u so s´anh I v`a chuˆo˜i so s´anh 1) ta r´ut ra:

Dˆa´u hiˆe.u D’Alembert. Nˆe´u chuˆo˜i a1+a2+· · ·+an+ ., an >0

∀n c´o

lim n→∞

an+1

an =D

th`ı chuˆo˜i hˆo.i tu khiD<1 v`a phˆan k`y D>1

Dˆa´u hiˆe.u Cauchy. Nˆe´u chuˆo˜i a1+a2 +· · ·+an+ ., an >0 ∀n

c´o

lim n→∞

n

√

an =C

th`ı chuˆo˜i hˆo.i tu khiC <1 v`a phˆan k`y C >1

Trong tru.`o.ng ho..p D = C = th`ı ca’ hai dˆa´u hiˆe.u n`ay dˆe`u khˆong cho cˆau tra’ l`o.i kh˘a’ng di.nh v`ı tˆo` n ta.i chuˆo˜i hˆo.i tu lˆa˜n chuˆo˜i phˆan k`y v´o.i D ho˘a.cC b˘a`ng

Dˆa´u hiˆe.u t´ıch phˆan. Nˆe´u h`am f(x) x´ac di.nh ∀x > khˆong ˆam v`a gia’m th`ı chuˆo˜i P

n>1

∞

Z

0

f(x)dx hˆo.i tu

T`u dˆa´u hiˆe.u t´ıch phˆan suy chuˆo˜i P n>1

1

nα hˆo.i tu α > v`a phˆan k`y 0< α61 Nˆe´u α60 th`ı an=

nα 6→0 α60 v`a n→ ∞nˆen chuˆo˜i d˜a cho c˜ung phˆan k`y

C ´AC V´I DU. V´ı du 1. Kha’o s´at su. hˆo.i tu cu’a c´ac chuˆo˜i

1) X

n>1

1

p

n(n+ 1); 2)

X

n>7

1 nlnn·

Gia’i. 1) Su.’ du.ng bˆa´t d˘a’ng th´u.c hiˆe’n nhiˆen

p

n(n+ 1) > n+ · V`ı chuˆo˜i P

n>1

1

n+ l`a phˆ` n du sau sˆo´ ha.ng th´u nhˆa´t cu’a chuˆo˜i diˆe`ua h`oa nˆen n´o phˆan k`y

Do d´o theo dˆa´u hiˆe.u so s´anh I chuˆo˜i d˜a cho phˆan k`y. 2) V`ı lnn >2∀n > nˆen

nlnn <

n2 ∀n >7

Do chuˆo˜i Dirichlet P n>7

1

n2 hˆo.i tu nˆen suy r˘a`ng chuˆo˜i d˜a cho hˆo.i

tu N

V´ı du 2. Kha’o s´at su. hˆo.i tu cu’a c´ac chuˆo˜i:

1) X

n>1

(n−1)n

nn+1 , 2)

X

n>1

n2e− √

n .

Gia’i. 1) Ta viˆe´t sˆo´ ha.ng tˆo’ng qu´at cu’a c´ac chuˆo˜i du.´o.i da.ng: (n−1)n

nn+1 =

1 n

1−

n

Ta biˆe´t r˘a`ng lim n→∞

1−

n

n =

e nˆenann→∞∼ ne Nhu.ng chuˆo˜i P

n→∞

ne phˆan k`y, d´o chuˆo˜i d˜a cho phˆan k`y 2) R˜o r`ang l`a dˆa´u hiˆe.u D’Alembert v`a Cauchy khˆong gia’i quyˆe´t du.o c vˆa´n dˆe` vˆe` su hˆo.i tu Ta nhˆa.n x´et r˘a`nge−

√ n

= 0(n−α2) khin→ ∞

(α >0) T`u d´o

X

n>1

an=X n>1

1 na20−2

hˆo.i tu nˆe´u a0 >6 Do vˆa.y theo dˆa´u hiˆe.u so s´anh I chuˆo˜i

P

n>1

n2e−√n hˆo.i tu N

V´ı du 3. Kha’o s´at su hˆo.i tu cu’a chuˆo˜i

1) X

n>1

2n+n2

3n+n , 2)

X

n>1

(n!)2

(2n)!·

Gia’i. 1) Ta c´o: an+1

an =

2n+1+ (n+ 1)2

3n+1+ (n+ 1) ×

3n+n 2n+n2 =

2 + (n+ 1)

2

2n

3 +n+ 3n

×

1 + n 3n + n

2

2n ,

n

√

an=

n

v u u u u t

1 +n

2

2n + n

3n

·

T`u d´o suy lim n→∞

an+1

an =

3 v`a limn→∞

n

√

an =

3 V`a ca’ hai dˆa´u hiˆe.u Cauchy, D’Alembert dˆ`u cho kˆe´t luˆa.n chuˆo˜i hˆo.i tu e

2) ´Ap du.ng dˆa´u hiˆe.u D’Alembert ta c´o:

D= lim n→∞

an+1

an = limn→∞

(n+ 1)2

Nhˆa n x´et. Nˆe´u ´ap du.ng bˆa´t d˘a’ng th´u.c

n

e

n

< n!< e

n

2

n th`ı

(n!)n2

(2n)!

1

n

< en2

n

2

2

2n

e

2 =

e2+2n

42 ,

do d´o lim n→∞

n

√

an <

e

4

2

< v`a d´o dˆa´u hiˆe.u Cauchy c˜ung cho ta kˆe´t luˆa.n.

V´ı du 4. Kha’o s´at su hˆo.i tu cu’a chuˆo˜i

1) X

n>1

2n

n2+ 1, 2)

X

n>2

1

nlnpn, p >

Gia’i. 1) Ta c´o an= 2n

n2+ 1 =f(n) Trong biˆe’u th´u.c cu’a sˆo´ ha.ng

tˆo’ng qu´at cu’a an = 2n

n2+ 1 ta thay n bo.’ i biˆe´n liˆen tu.c x v`a ch´u.ng to’

r˘a`ng h`am f(x) thu du.o c liˆen tu.c do.n diˆe.u gia’m trˆen nu.’a tru.c du.o.ng. Ta c´o:

+∞

Z

1

2x

x2+ 1dx= limA→+∞

A

Z

1

2x

x2+ 1dx= limA→+∞ln(x

+ 1)A1 = ln(+∞)−ln =∞.

Do d´o chuˆo˜i 1) phˆan k`y

2) Nhu trˆen, ta d˘a.t f(x) =

xlnpx, p > 0,x >2 H`am f(x) tho’a m˜an mo.i diˆe`u kiˆe.n cu’a dˆa´u hiˆe.u t´ıch phˆan V`ı t´ıch phˆan

+∞

Z

2

V´ı du 5. Ch´u.ng minh r˘a`ng chuˆo˜i P n>1

n+

(n+ 1)√n tho’a m˜an diˆ`u kiˆe.ne cˆ` n hˆo.i tu nhu.ng chuˆo˜i phˆan k`y.a

Gia’i. Ta c´o

an= n+

(n+ 1)√n (n→∞∼ )

1

√

n ⇒nlim→∞an= Tiˆe´p theo∀k = 1,2, , n ta c´o

ak = k+ (k+ 1)

√

k >

√

k >

√

n v`a d´o

sn = n

X

k=1

ak >n·√1

n =

√

n →+∞ khin → ∞

v`a d´o chuˆo˜i phˆan k`y N

B `AI T ˆA P

Trong c´ac b`ai to´an sau dˆay, b˘a`ng c´ach kha’o s´at gi´o.i ha.n cu’a tˆo’ng riˆeng, h˜ay x´ac lˆa.p t´ınh hˆo.i tu (v`a t´ınh tˆo’ngS) hay phˆan k`y cu’a chuˆo˜i

1. X n>1

1

3n−1 (DS S =

3 2) 2. X

n>0

(−1)n

2n (DS 3) 3. X

n>1

(−1)n−1 (DS Phˆan k`y) 4. X

n>0

ln2n2 (DS 1−ln22) 5. X

n>1

1

6. X n>1

1

(α+n)(α+n+ 1), α>0 (DS α+ 1) 7. X

n>3

1

n2−4 (DS

25 48) 8. X

n>1

2n+

n2(n+ 1)2 (DS 1)

9. X n>1

(√3

n+ 2−1√3

n+ +√3

n). (DS 1−√3

2)

10. X

n>1

1

n(n+ 3)(n+ 6) (DS 73 1080)

Su.’ du.ng diˆe`u kiˆe.n cˆa` n 2) dˆe’ x´ac di.nh xem c´ac chuˆo˜i sau dˆay chuˆo˜i n`ao phˆan k`y

11. X

n>1

(−1)n−1 (DS Phˆan k`y)

12. X

n>1

2n−1

3n+ (DS Phˆan k`y)

13. X

n>1

n

p

0,001 (DS Phˆan k`y)

14. X

n>1

1

√

2n (DS Dˆa´u hiˆe.u cˆa` n khˆong cho cˆau tra’ l`o.i)

15. X

n>1

2n

3n (DS Dˆa´u hiˆe.u cˆa` n khˆong cho cˆau tra’ l`o.i)

16. X

n>1

1

n

√

0,3 (DS Phˆan k`y)

17. X

n>1

1

n

√

n! (DS Phˆan k`y)

18. X

n>1

n2sin

19. X n>1

1 + n

n2

en (DS Phˆan k`y)

20. X

n>1

2n2+ 1

2n2+ 3

n2

(DS Phˆan k`y)

21. X

n>1

nn+n1

n+ n

n (DS Phˆan k`y)

22. X

n>1

n+

(n+ 1)√n (DS Dˆa´u hiˆe.u cˆa` n khˆong cho cˆau tra’ l`o.i)

23. X

n>1

(n+ 1)arctg

n+ (DS Phˆan k`y)

Trong c´ac b`ai to´an sau dˆay, h˜ay d`ung dˆa´u hiˆe.u so s´anh dˆe’ kha’o s´at su hˆo.i tu cu’a c´ac chuˆo˜i d˜a cho

24 X

n>1

1

√

n (DS Phˆan k`y)

25. X

n>1

1

nn (DS Hˆo.i tu.) Chı’ dˆa˜n. n

n>2n ∀n>3.

26. X

n>1

1

lnn (DS Phˆan k`y) Chı’ dˆa˜n. So s´anh v´o.i chuˆo˜i diˆe`u h`oa

27. X

n>1

1

n3n−1 (DS Hˆo.i tu.)

28. X

n>1

1

3

√

n+ (DS Phˆan k`y)

29. X

n>1

1

2n+ 1 (DS Hˆo.i tu.)

30. X

n>1

n

31. X n>1

1

p

(n+ 2)(n2 + 1) (DS Hˆo.i tu.)

32. X

n>1

5n2−3n+ 10

3n5+ 2n+ 17 (DS Hˆo.i tu.)

33. X

n>1

5 + 3(−1)n

2n+3 (DS Hˆo.i tu.) Chı’ dˆa˜n. 265 + 3(−1)

n 68.

34. X

n>1

lnn

n (DS Phˆan k`y) Chı’ dˆa˜n. lnn >1 ∀n >2

35. X

n>1

lnn

n2 (DS Hˆo.i tu.)

Chı’ dˆa˜n. Su.’ du.ng hˆe th´u.c lnn < nα ∀α >0 v`an du’ l´o.n

36. X

n>1

lnn

3

√

n (DS Phˆan k`y)

37. X

n>1

n5

5√n (DS Hˆo.i tu.)

38. X

n>1

1

√

nsin

n (DS Hˆo.i tu.)

39. X

n>1

n4+ 4n2+

2n (DS Hˆo.i tu.)

40. X

n>1

n2(√n

a− n+1√

a), a >0 (DS Phˆan k`y ∀a6= 1)

41. X

n>1

(n

√

2− n+1 √

2) (DS Hˆo.i tu.)

42. X

n>1

1

1 +an,a >0 (DS Hˆo.i tu khia >1 Phˆan k`y < a61)

43. X

n>1

sin πn

n2√n+n+ 1 (DS Hˆo.i tu.)

44. X n>1

sinπ n

p

, p > (DS Hˆo.i tu nˆe´u p > 1, phˆan k`y nˆe´u p61)

45. X

n>1

tgp π

n+ 2, p >0 (DS Hˆo.i tu p >1, phˆan k`y p61)

46. X

n>1

sin np ·tg

1

nq, p >0, q >0

(DS Hˆo.i tu khi p+q >1, phˆan k`y p+q 61)

47. X

n>1

1−cos np

, p >0

(DS Hˆo.i tu khi p >

2, phˆan k`y khip6 2)

48. X

n>1

(

√

n+ 1−√n)pln2n+ 2n+

(DS Hˆo.i tu khi p >0, phˆan k`y p60)

Trong c´ac b`ai to´an sau dˆay, h˜ay kha’o s´at su hˆo.i tu cu’a chuˆo˜i d˜a cho nh`o dˆa´u hiˆe.u du’ D’Alembert

49. X

n>1

n

2n (DS Hˆo.i tu.)

50. X

n>1

2n−1

nn (DS Hˆo.i tu.)

51. X

n>1

2n−1

(n−1)! (DS Hˆo.i tu.)

52. X

n>1

n!

2n+ 1 (DS Phˆan k`y)

53. X

n>1

4nn!

nn (DS Phˆan k`y)

54. X

n>1

3n

55. X n>1

1·3· · ·(2n−1)

3nn! (DS Hˆo.i tu.)

56. X

n>1

n2sin π

2n (DS Hˆo.i tu.)

57. X

n>1

n(n+ 1)

3n (DS Hˆo.i tu.)

58. X

n>1

73n

(2n−5)! (DS Hˆo.i tu.)

59. X

n>1

(n+ 1)!

2nn! (DS Hˆo.i tu.)

60. X

n>1

(2n−1)!!

n! (DS Phˆan k`y)

61. X

n>1

n!(2n+ 1)!

(3n)! (DS Hˆo.i tu.)

62. X

n>1

nnsin π 2n

n! (DS Phˆan k`y)

63. X

n>1

nn

n!3n (DS Hˆo.i tu.)

64. X

n>1

n!an

nn ,a6=e,a >0 (DS Hˆo.i tu khia < e, phˆan k`y khia > e) Trong c´ac b`ai to´an sau dˆay, h˜ay kha’o s´at su. hˆo.i tu cu’a chuˆo˜i d˜a cho nh`o dˆa´u hiˆe.u du’ Cauchy

65. X

n>1

n

2n+

n

(DS Hˆo.i tu.)

66. X

n>1

arc sin1 n

n

(DS hˆo.i tu.)

67. X

n>1

1 3n

n+ 1

n

n2

68. X n>1

n53n+ 4n+

n

(DS Hˆo.i tu.)

69. X

n>1

3n

n+

nn+ 2

n+

n2

(DS Phˆan k`y)

70. X

n>1

n!

n√n (DS Phˆan k`y)

Chı’ dˆan. Su.’ du.ng cˆong th´u.c Stirlingn!∼

n

e

n√

2πn, n → ∞

71. X

n>1

n−1

n+

n(n−1)

(DS Hˆo.i tu.)

72. X

n>1

n2

+ n2+ 4

n3+1

(DS Hˆo.i tu.)

73. X

n>1

3n n n+

n2

(DS Phˆan k`y)

74. X

n>10

arctgn

√

3n+

√

n+ (DS Phˆan k`y)

75. X

n>1

an

n+

n

, a >0

(DS Hˆo.i tu 0 < a <1, phˆan k`y a>1)

76. X

n>1

nα

ln(n+ 1)n/2

, α >0 (DS Hˆo.i tu. ∀α)

77. X

n>1

5 + (−1)n

4n+1 (DS Hˆo.i tu.)

78. X

n>1

2(−1)n+n (DS Phˆan k`y)

79. X

n>1

2(−1)n−n (DS Hˆo.i tu.)

80. X

n>1

[5−(−1)n]n

81. X n>1

[5 + (−1)n]n

n27n (DS Hˆo.i tu.)

82. X

n>1

[3 + (−1)n]

3n (DS Hˆo.i tu.)

83. X

n>1

n4[√5 + (−1)n]n

4n (DS Hˆo.i tu.)

84. X

n>1

2 + (−1)n

5 + (−1)n+1 · (DS Hˆo.i tu.)

13.2 Chuˆo˜i hˆo.i tu tuyˆe.t dˆo´i v`a hˆo.i tu. khˆong tuyˆe.t dˆo´i

13.2.1 C´ac di.nh ngh˜ıa co ba’n Chuˆo˜i v´o.i c´ac sˆo´ ha.ng c´o dˆa´u kh´ac

a1+a2+· · ·+an+· · ·=

X

n>1

an (13.2)

du.o c go.i l`a chuˆo˜i hˆo.i tu tuyˆe.t dˆo´i nˆe´u chuˆo˜i sˆo´ du.o.ng

|a1|+|a2|+· · ·+|an|+· · ·=

X

n>1

|an| (13.3) hˆo.i tu Chuˆo˜i (13.2) du.o c go.i l`a chuˆo˜ihˆo i tu c´o diˆ`u kiˆe.n (khˆong tuyˆe.te dˆo´i) nˆe´u n´o hˆo.i tu c`on chuˆo˜i (13.3) phˆan k`y.

D- i.nh l´y 13.2.1. Mo i chuˆo˜i hˆo.i tu tuyˆe.t dˆo´i dˆe`u hˆo.i tu., t´u.c l`a su hˆo.i tu cu’a chuˆo˜i (13.3) k´eo theo su hˆo.i tu cu’a chuˆo˜i (13.2)

13.2.2 Chuˆo˜i dan dˆa´u v`a dˆa´u hiˆe.u Leibnitz Chuˆo˜i da.ng

X

n>1

(−1)n−1an=a1−a2+a3−a4+· · ·+ (−1)n−1an+ ,

an>0∀n ∈N (13.4)

du.o c go.i l`a chuˆo˜i dan dˆa´u

Dˆa´u hiˆe.u Leibnitz. Nˆe´u lim

n→∞an = v`a an >an+1 >0

∀n ∈N th`ı chuˆo˜i dan dˆa´u (13.4) hˆo.i tu v`a

|S−Sn|6an+1 (13.5)

trong d´o S l`a tˆo’ng cu’a chuˆo˜i (13.4), Sn l`a tˆo’ng riˆeng th´u.n cu’a n´o Nhu vˆa.y dˆe’ kha’o s´at su hˆo.i tu cu’a chuˆo˜i dan dˆa´u ta cˆa` n kiˆe’m tra hai diˆ`u kiˆe.ne

i)an>an+1 >0 ∀n∈N,

ii) lim

n→∞an=

Hˆe th´u.c (13.5) ch´u.ng to’ r˘a`ng sai sˆo´ g˘a.p pha’i thay tˆo’ng S cu’a chuˆo˜i dan dˆa´u hˆo.i tu bo.’i tˆo’ng cu’a mˆo.t sˆo´ sˆo´ ha.ng dˆa` u tiˆen cu’a n´o l`a khˆong vu.o t qu´a gi´a tri tuyˆe.t dˆo´i cu’a sˆo´ ha.ng th´u nhˆa´t cu’a chuˆo˜i du. bi c˘a´t bo’

Dˆe’x´ac lˆa p su hˆo.i tu.cu’a chuˆo˜i v´o.i c´ac sˆo´ ha.ng c´o dˆa´u kh´ac ta c´o thˆe’ su.’ du.ng c´ac dˆa´u hiˆe.u hˆo.i tu cu’a chuˆo˜i du.o.ng v`a di.nh l´y 13.1.1. Nˆe´u chuˆo˜i P

n>1

|an| phˆan k`y th`ı su hˆo.i tu cu’a chuˆo˜i P n>1

an tro.’ th`anh vˆa´n dˆ` dˆe’ mo.e ’ ngoa.i tr`u tru.`o.ng ho p su.’ du.ng dˆa´u hiˆe.u D’Alembert v`a dˆa´u hiˆe.u Cauchy v`ı c´ac dˆa´u hiˆe.u n`ay x´ac lˆa.p su phˆan k`y cu’a chuˆo˜i chı’ du a trˆen su ph´a v˜o diˆe`u kiˆe.n cˆa` n

Nhˆa n x´et. Chuˆo˜i dan dˆa´u tho’a m˜an dˆa´u hiˆe.u Leibnitz go.i l`a chuˆo˜i Leibnitz

V´ı du 1. Kha’o s´at su hˆo.i tu v`a d˘a.c t´ınh hˆo.i tu cu’a chuˆo˜i P n>1

(−1)n−1

√

n

Gia’i. D˜ay sˆo´

1

√

n

do.n diˆe.u gia’m dˆa` n dˆe´n n→ ∞ Do d´o theo dˆa´u hiˆe.u Leibnitz n´o hˆo.i tu Dˆe’ kha’o s´at d˘a.c t´ınh hˆo.i tu (tuyˆe.t dˆo´i hay khˆong tuyˆe.t dˆo´i) ta x´et chuˆo˜i du.o.ng P

n>1

1

√

n Chuˆo˜i n`ay phˆan k`y Do vˆa.y chuˆo˜i d˜a cho hˆo.i tu c´o diˆe`u kiˆe.n. N

V´ı du 2. Kha’o s´at su. hˆo.i tu v`a d˘a.c t´ınh hˆo.i tu cu’a chuˆo˜i

X

n>1

(−1)n−1ln

2

n n ·

Gia’i. Dˆe’ kha’o s´at d´ang diˆe.u cu’a d˜ay

ln2

n n

ta x´et h`am ϕ(x) = ln2x

x R˜o r`ang l`a limx→∞ϕ(x) = v`a ϕ

0(x) = lnx

x2 (2−lnx) T`u d´o suy

ra x > e2 th`ı ϕ0(x) < 0 Do d´o d˜ay (an) = ln

2

n

n tho’a m˜an dˆa´u hiˆe.u Leibnitz v´o.in > e2 V`ı vˆ

a.y chuˆo˜i d˜a cho hˆo.i tu Dˆe˜ d`ang thˆa´y r˘a`ng chuˆo˜i sˆo´ du.o.ng P

n>1

ln2n

n phˆan k`y nˆen chuˆo˜i dan dˆa´u d˜a cho hˆo.i tu c´o diˆe`u kiˆe.n N

V´ı du 3. C˜ung ho’i nhu trˆen v´o.i chuˆo˜i

X

n>1

cosnα 2n ·

Gia’i. Dˆay l`a chuˆo˜i dˆo’i dˆa´u X´et chuˆo˜i du.o.ng

X

n>1

|cosnα|

2n (*)

V`ı|cosαn| 2n

1

V´ı du 4. C˜ung ho’i nhu trˆen dˆo´i v´o.i chuˆo˜i

X

n>1

(−1)n n(n+ 1)·

Gia’i. Dˆ˜ d`ang thˆa´y r˘a`ng d˜aye

n(n+ 1) do.n diˆe.u gia’m dˆa` n dˆe´n khin → ∞ Do d´o theo dˆa´u hiˆe.u Leibnitz n´o hˆo.i tu Ta x´et su hˆo.i tu. cu’a chuˆo˜i du.o.ng P

n>1

1

n(n+ 1) Chuˆo˜i n`ay hˆo.i tu., ch˘a’ng ha.n theo dˆa´u hiˆe.u t´ıch phˆan

∞

Z

1

dx

x(x+ 1) = limA→∞ A

Z

1

dx+

x+1

2

−

4

= lim A→∞ln

x x+

A

1

= ln

Do d´o chuˆo˜i d˜a cho hˆo.i tu tuyˆe.t dˆo´i. N

V´ı du 5. Cˆ` n lˆa´y bao nhiˆeu sˆo´ ha.ng cu’a chuˆo˜ia P n>1

(−1)n−1

n2 dˆe’ tˆo’ng

cu’a ch´ung sai kh´ac v´o.i tˆo’ng cu’a chuˆo˜i d˜a cho khˆong qu´a 0,01 ? 0,001 ?

Gia’i. 1+ Chuˆo˜i d˜a cho l`a chuˆo˜i Leibnitz Do d´o phˆa` n du cu’a n´o tho’a m˜an diˆ`u kiˆe.ne

|Rn|< an+1 ⇒ |Rn| <

1 (n+ 1)2 ·

Dˆe’ t´ınh tˆo’ng cu’a chuˆo˜i d˜a cho v´o.i su sai kh´ac khˆong qu´a 0,01 ta cˆa` n d`oi ho’i l`a

|Rn|<0,01 ⇒

(n+ 1)2 <0,01⇔n > 10

Nhu vˆa.y dˆe’ t´ınh tˆo’ng cu’a chuˆo˜i v´o.i sai sˆo´ khˆong vu.o t qu´a 0,01 ta chı’ cˆ` n t´ınh tˆo’ng mu.`o.i sˆo´ ha.ng dˆaa ` u l`a du’

Nhˆa n x´et. Ta thˆa´y r˘a`ng chuˆo˜i Leibnitz l`a cˆong cu t´ınh to´an tiˆe.n ho.n so v´o.i chuˆo˜i du.o.ng Ch˘a’ng ha.n dˆe’ t´ınh tˆo’ng cu’a chuˆo˜i P

n>1

1 n2

v´o.i sai sˆo´ khˆong vu.o..t qu´a 0,001 ta cˆa` n pha’i lˆa´y 1001 sˆo´ ha.ng m´o.i du’. Thˆa.t vˆa.y ta c´o thˆe’ ´ap du.ng dˆa´u hiˆe.u t´ıch phˆan Ta c´o

∞

Z

n+1

f(x)dx < Rn< ∞

Z

n

f(x)dx.

T`u d´o

Rn< ∞

Z

n dx x2 =−

1 x

∞ n

= n· T`ımndˆe’

n <0,001 Gia’i bˆa´t phu.o.ng tr`ınh dˆo´i v´o.in ta c´on >1000, t´u.c l`a R1001 < 0,001 Vˆa.y ta cˆa` n lˆa´y 1001 sˆo´ ha.ng dˆa` u dˆe’ t´ınh tˆo’ng

m´o.i c´o du.o..c sai sˆo´ khˆong qu´a 0,001 V´ı du 6. Ch´u.ng to’ r˘a`ng chuˆo˜i

2 +5 −

7

+10 −

26 27

+· · ·+n

2

+ n2 −

n3−1 n3

+ . (*) hˆo.i tu., c`on chuˆo˜i

2 + −

7 +

10 −

26

27 +· · ·+

n2+ n2 −

n3 −1

n3 + . (**)

thu du.o c t`u chuˆo˜i d˜a cho sau bo’ c´ac dˆa´u ngo˘a.c do.n l`a chuˆo˜i phˆan k`y

Gia’i. Sˆo´ ha.ng tˆo’ng qu´at cu’a chuˆo˜i (*) c´o da.ng an = n

2+ 1

n2 −

n3−1

n3 =

n+ n3 ·

Do d´o ∀n >1 ta c´o

n+ n3 =

1 n2 +

v`a chuˆo˜i P n>1

1

nα hˆo.i tu ∀α >1 nˆen chuˆo˜i d˜a cho hˆo.i tu

Bˆay gi`o x´et chuˆo˜i (**) R˜o r`ang sˆo´ ha.ng tˆo’ng qu´at cu’a (**) khˆong dˆ` n dˆe´n khia n → ∞, d´o chuˆo˜i (**) phˆan k`y N

B `AI T ˆA P

Su.’ du.ng dˆa´u hiˆe.u Leibnitz dˆe’ ch´u.ng minh c´ac chuˆo˜i sau dˆay hˆo.i tu c´o diˆe`u kiˆe.n

1. X n>4

(−1)n+1

√

n2−4n+ 1

2. X n>1

(−1)n+1n9

√

n20+ 4n3+ 1

3. X n>1

(−1)nn (n+ 1)√3

n+ 4. X

n>1

(−1)n√n n+ 20 5. X

n>1

(−1)n

4

√

n

6. X n>1

(−1)nlnn n 7. X

n>1

(−1)n+1 2n+ n(n+ 1)

8. X n>1

(−1)ncos π n n 9. X

n>1

(−1)n(n

√

2−1)

10. X

n>1

(−1)nn−1 n+

1

100√n

Kha’o s´at su hˆo.i tu v`a d˘a.c t´ınh hˆo.i tu cu’a c´ac chuˆo˜i

11. X

n>1

(−1)n2n+ 3n−2

n

(DS Hˆo.i tu tuyˆe.t dˆo´i)

12. X

n>1

(−1)n

3n+ 1

3n−2

5n+2

(DS Phˆan k`y)

13. X

n>1

(−1)n2 + (−1) n

n (DS Phˆan k`y)

14. X

n>1

(−1)n−1 n sin

√

n

15. X n>1

(−1)n+1arctgln(n+ 1)

(n+ 1)2 (DS Hˆo.i tu tuyˆe.t dˆo´i)

Chı’ dˆa˜n. Su.’ du.ng bˆa´t d˘a’ng th´u.c ln(n+ 1)<√n+ 1,n >2

16. X

n>1

(−1)n+1

n−ln3n (DS Hˆo.i tu c´o diˆe`u kiˆe.n)

Trong c´ac b`ai to´an sau dˆay h˜ay x´ac di.nh gi´a tri cu’a tham sˆo´p dˆe’ chuˆo˜i sˆo´ hˆo.i tu tuyˆe.t dˆo´i ho˘a.c hˆo.i tu c´o diˆe`u kiˆe.n

17. X

n>1

(−1)n−1

(2n−1)p, p >0

(DS Hˆo.i tu tuyˆe.t dˆo´i khip >1; hˆo.i tu c´o diˆe`u kiˆe.n 0 < p61)

18. X

n>1

(−1)n−1tgp

n√n, p >0 (DS Hˆo.i tu tuyˆe.t dˆo´i khip >

3; hˆo.i tu c´o diˆe`u kiˆe.n 0< p 3)

19. X

n>1

(−1)n−1sinp 5n+

n2√n+ 3,p >0

(DS Hˆo.i tu tuyˆe.t dˆo´i khip >

3; hˆo.i tu c´o diˆe`u kiˆe.n 0< p 3)

20. X

n>1

(−1)n−1

√

n

lnn+ n+

p

, p >0

(DS Hˆo.i tu tuyˆe.t dˆo´i khip >

2; hˆo.i tu c´o diˆe`u kiˆe.n 0< p 2) Kha’o s´at d˘a.c t´ınh hˆo.i tu cu’a c´ac chuˆo˜i (21-32):

21. X

n>1

(−1)n+1

n√3 n (DS Hˆo.i tu tuyˆe.t dˆo´i)

22. X

n>1

(−1)n+1

23. X n>1

(−1)n−1 (2n+ 1)!!

2·5·8· · ·(3n−1) (DS Hˆo.i tu tuyˆe.t dˆo´i)

24. X

n>1

(−1)n+11−cos√π

n

(DS Hˆo.i tu c´o diˆe`u kiˆe.n)

25. X

n>1

(−1)nsinπ n

n (DS Hˆo.i tu tuyˆe.t dˆo´i)

26. X

n>1

(−1)n

√

n+ (DS Hˆo.i tu c´o diˆe`u kiˆe.n)

27. X

n>1

(−1)n

n

√

n (DS Phˆan k`y)

28. X

n>1

(−1)n+1

n−lnn (DS Hˆo.i tu c´o diˆe`u kiˆe.n)

29. X

n>1

(−1)n−1

(n+ 1)a2n

(DS Hˆo.i tu tuyˆe.t dˆo´i khi |a| >1, hˆo.i tu c´o diˆe`u kiˆe.n khi |a|= 1, phˆan k`y |a|<1)

30. X

n>1

(−1)n

(n+ 1)(√n+ 1−1) (DS Hˆo.i tu tuyˆe.t dˆo´i)

31. X

n>1

(−1)n+12 +

n

n

5n (DS Hˆo.i tu tuyˆe.t dˆo´i)

32. X

n>1

(−1)ntg π

3n (DS Hˆo.i tu tuyˆe.t dˆo´i)

Trong c´ac b`ai to´an sau dˆay, h˜ay t`ım sˆo´ sˆo´ ha.ng cu’a chuˆo˜i d˜a cho cˆ` n lˆa´y dˆe’ tˆo’ng cu’a ch´a ung v`a tˆo’ng cu’a chuˆo˜i tu.o.ng ´u.ng sai kh´ac mˆo.t da.i lu.o ng khˆong vu.o t qu´a sˆo´δ cho tru.´o.c

33. X

n>1

(−1)n−1

34. X n>1

cos(nπ)

n! ,δ = 0,001 (DS N o= 5)

35. X

n>1

(−1)n−1

√

n2+ 1,δ = 10

−6. (DS. N o= 106)

36. X

n>1

cosnπ

2n(n+ 1), δ= 10

−6. (DS. N o= 15)

37. X

n>1

(−1)n2n

(4n+ 1)5n, δ= 0,1?; δ = 0,01? (DS.N o = 2, N o= 3)

38. X

n>1

(−1)n

n! , δ= 0,1;δ = 0,001? (DS.N o = 4, N o= 6)

13.3 Chuˆo˜i l˜uy th`u.a 13.3.1 C´ac di.nh ngh˜ıa co ba’n

Chuˆo˜i l˜uy th`u.a dˆo´i v´o.i biˆe´n thu c x l`a chuˆo˜i da.ng

X

n>0

anxn =a0+a1x+a2x2+· · ·+anxn+ . (13.6)

hay

X

n>0

an(x−a)n=a0+a1(x−a) +· · ·+an(x−a)

n

+ . (13.7)

trong d´o c´ac hˆe sˆo´a0, a1, , an, l`a nh˜u.ng h˘a`ng sˆo´ B˘a`ng ph´ep dˆo’i

biˆe´n x bo.’ i x−a t`u (13.6) thu du.o c (13.7) Do d´o dˆe’ tiˆe.n tr`ınh b`ay ta chı’ cˆ` n x´et (13.6) l`a du’ (t´a u.c l`a xem a= 0)

Chuˆo˜i (13.6) luˆon hˆo.i tu ta.i diˆe’mx= 0, c`on (13.7) hˆo.i tu ta.ix=a. Do d´o tˆa.p ho p diˆe’m m`a chuˆo˜i l˜uy th`u.a hˆo.i tu luˆon luˆon 6=∅