Xac djnh nguyen ham bang vi?c suf dung bang cac nguyen ham ca ban

Bai 6. Tim nghipm tren nufa khoang cua phiTdng trinh

1. Xac djnh nguyen ham bang vi?c suf dung bang cac nguyen ham ca ban

Bai 1. Tinh cac nguyen ham sau

a. j(2x + 3)-^dx b. jcos'*x .sinxdx c.

2e' e ^ + 1

-dx d. K21nx + i r dx

a. Ta c6: f(2x + 3)-^dx = 1 f(2x + 3)-^d(2x + 3) = i ^ ^ ^ ^ c = ^-^^ + C 2 J 2 4 8 b. Ta c6: cos"* x.sinxdx = -fees'* xd(cosx) = —cos^x + C

J J 5 , Taco: f - ^ d x =^2 f ^^^^SlilA = 21n(e^ . 1) + C

- , , •'e' +1 e' +1

d. Taco: J-(2 In x + 0_ = 1 j(2 In X +1)2 d(2 In x +1) = 1(2 In x + 1 ) ' + C Bai 2. Tinh cac nguyen ham sau

a. J 2 s i n 2 ^ x b. Jcot^ xdx c.

Hifdng dan giai a. Tac6: 2sin2 ^ x

tan xdx d. [ dx tanx

• COS"* X

( l - c o s x ) d x = x - s i n x + C , j , h.

c.

col xdx = J 1 |dx = - c o t x - x + C '! i • sm X J

tan , fsmx , fd(cosx) , _

xdx=: I dx = - \ ^ = - l n cosx +C cos X '•"'^ ^

fd(cosx) cosx

COng ty TNHH MTV D i / / Khan,: l '

f t a n x , r sinx , fd(cosx) 1 _3 _

— - — d x = T —d x = - | 7— = -cos ^x + C =

cos X f f-ns X 3

d. Taco:

pal 3- Tinh cac nguyen ham sau:

1 cos' X 3

• J '

3cos^ X + C

a. -dx b . j . 1

x^ - 3 x + 2 -dx

a. Ta c6:

1 + x'

Hifdng dSn giai

^ ^ ' ' ^ ^ ^ ^ ' ) = : l l n ( l . x 2 ) . C i + x '

dx

= l n x- 2- l n x - l + C = ln x- 2 x - 1 + C 2. Xac djnh nguy&n ham b^ng phi/ofng phap phan tich

PhUdng phap phan tich thifc chat la vi<;c suT dung cac dong nhat thtfc dc bien ddi bieu thufc di/(3i da'u tich phan lhanh long cac bieu thiJc ma nguyen ham cua moi bieu Ihtfc do c6 the nhan diTcJc tiT bang nguyen ham hoSc chi bling cac phep bien ddi ddn gian da bie't.

Chiiy: Diem ma'u chdt cua phep phan tich la co the rut ra y tirdng rieng cho minh tif mot vai minh hpa sau: =: i atvi

• Vdi f(x) = ( x ' - if ihi Viet lai f(x) = x*" - 4x' + 4

• Vdi f(x) = ^^""^^/-^ thi Viet lai i(x) = x - 3 + ^ Vdi ITx) =

x - 1

1 thi Viet lai f(x) = 1

x - 1 1 x - 3 x- 2 x^ -5x-(-6

• Vdi f(x) = ^ , thi Viet lai f(x) = l(V2x + 3 - V2x + 1)

72X + 1+V2X + 3 2^ '

• Vdi f(x) = (2* - 3Y thi Viet lai f(x) = 4" - 2.6* + 9'

• Vdi f(x) = 8.cos\.sinx thi vic't lai , f(x) = 2(cos3x + 3cosx).sinx = 2cos3x.sinx + 6cosx.sinx = sin4x + 2sin2x

Bai 1. Tinh nguyen hSm I = fx(l-x)2"*"'dx Hifdng dSn giai

• tan^x = ( l + t a n M - 1

• cot^x = ( l + c o t ^ x ) - 1

Str dung dong nhat thi?c: x = 1 - (1 - x), ta dU'dc x ( l

Khi do

x ( l - x f " " = [ I - (1 - x)](l - x ^ " = (1 - x f - ' " - (1 - x)

Luyfn gUii tru6c ky thi Dff 3 miin Bdc, Trung, Nam Todn hoc - NguySn Van ThOng

I = f(l - x ) 2 " " ô d x - J(l - x)2"'"^dx = - X)^"<"*d(l - X ) + J(l - x)2"<"^d(l - X )

2009 2010

* (Bai todn e/iia k/iffa: Tinh nguyen ham Jx(ax + b ) " d x v6ia=^0 ' ^1 Suf d u n g d o n g n h a t thiJc x = — .ax = — [(ax + b ) - b l , la du'rtc

a a

x ( a x + b ) " = - f ( a x + b ) - b ] ( a x + b ) " = -!-(ax + b ) " ^ ' - - ( a x + b ) "

a a a

T a x c t ba IrU'dng h d p " ' 1 .^.

• a = - 2 , ta di/dc I

ffiftp a = - 1 , ta dufdc I = 1 r

1

j( a x + b)~' d ( a x + b) - b j( a x + b)"^ d ( a x + b) In ax + b b

a

_ l_ r

„2

ax + b + C

1 r

|d(ax + b) - b J(ax + b ) " ' d ( a x + b)

>i.ff) rtfidfi •JiriJ i'vj hh 'iitdi i F ^ i f f i o r n BU'J

,2 L ax + b - b l n ax + b + C

"'US Y Bl Hlx j'i.u 1

a e R \ { - 2 ; - l ) , t a diTdc I = (ax + bf+^ Max + b)

a + 2 a + 1 Bai 2. Tinh nguyen ham I = J- dx

x ^ - 4 x + 3

+ c

Hi^dng dSn giai

Ta c6: 1 1

K h i do: I = - 2

1 ( x - l ) - ( x - 3 ) ^ 1 x ^ - 4 x + 3 ( x - 3 ) ( x - l ) 2 ( x - l ) ( x - 3 ) 2

d ( x - 3 ) f d ( x - l )

, x - 3 x - 1 r d x r a x dx

= - ( n | x - 3 | - l n | x - l | ) + C - - l n x - 3

x - 3 x - 1

x - 1 + C Bai 3. Tinh nguyen ham: I = f , ,

•'Vx + 2 - v x - ;

c d ' i 2. . ! V -HSx.:. H i ^ n g d S n g i a i

1 = - J(Vx + 2 + N/x-3)dx = i( | V x + 2d(x + 2 ) + j V x ^ d ( x- 3 ) ) 15L V(x + 2)-' + V ( x - 3 ) - ' + C ^ ' i *

pai 4- T i n h nguyen h a m : I = d x s m x . c o s X

H i ^ n g dSn giai \ t)(f.A u'. -'ip d,:C . 2 2

1 s m X + cos X sm X 1

- + • smx 2 1

Ta c6: ^— - 5 - T ' • ~ r

six.cos X s i n x . c o s x cos x s m x cos x ,.,,,,2 x x

L O S lan

Suy ra I = r s m x d x + ' c o s " X

- + l n

T X X

C O S " - .tan - 2 2

A - t

. f d ( C O S X ) r C O S " X

' x ^ Ian V 2 y

Ian

C O S X

tan — X

2 + C Bai 5. Tinh nguyen ham I = j -

Ta c6:

dx

X

•-tljSf.SS X V C O S X 4

Hif(}ng d i n giai

1= f — y — . — ^ d x = ^ f(l + tan^x)d(tanx)= fd(tanx)+ [tan^ x.d(lanx) cos X C O S X

= tanx + -lan"^x + C 3

U(A i'ii ằ<(*( ,

3. Xac djnh nguyen ham b^ng phi/dng phap tich phan tuTng phan u d v = u v -

* ^ i i todn cilia Idioa: Sfl" dung cong thiJc tich phan tifng phan:

Ta thi/c hien cac biTdc sau

Btf(?c 1 : Bien ddi tich phan ban dau ve dang: I = j f ( x ) d x = Jf,(x)f2(x)dx vdu

B i M c i i D a t u = f,(x) dv = f2(x)dx Bi|^c3: K h i do I = uv - fvdu

d u - f i ( x ) V = i;(x)dx

x l n x + Vx^ + 1

Bai 1. Tinh nguyen ham I = J ^ . ' dx

u = ln : + Vx^ + dv = x ^ + 1 .dx

HrfdngdSngiai ( ^ n ! 1 + X

du = —i^i==^ax = - 7 = .dx

v = Vx^ + l

93

•ằ'ằ&. n m' t J ằ J W ằ rằ v ô - J i a ằ 7 i . i i t u n J mjiu

= Vx^ +1 ln|x + Vx^ Jdx = >/x^+lln|x + V x ^ + l Bai 2. Tinh nguyen h^m I = Jcos(lnx)dx

H\idnf> dSn giai

- x + C

u = cos(lnx) dv ==dx

du = -—sin(lnx)dx

X V = X

I = xcos(lnx) + jsin(ln x)dx ..Lj,..:4:] (1)

, , , , , , i ( X "

I4 = Jsin(lnx)dx ftfci.

[ u = sin(lnx) I dv = dx

du = —cos(lnx)dx > * '0 1

X

v = x

• I , = x . s i n ( l n x ) - cos(lnx)dx = x s i n ( l n x ) - I (2) Ti/ (1) va (2) suy ra I = - [ c o s ( l n x ) + sin(ln x)] + C

Chii y: N c u bai loan ycu cau tinh gia trj cua c^p tich phan

Ii = |sin(ln x)dx va I = cos(in x)dx ihi ta ncn lifa chon each trinh bhy nhiTsau:

Suf dung tich phan tijfng phan cho I i D a t u = sin(lnx)

dv = dx

du - —cos(lnx)dx

X V = X

I| = x . s i n ( l n x ) - jcos(lnx)dx = x.sin(Inx)-l2 Suf dung tich phan tiTng phan cho I2

(3)

D S t u :=cos(lnx) dv = dx

du = - - s i n ( l n x ) d x

V = x

m&'

I2 = x . c o s ( l n x ) + sin(lnx)dx = xcos(lnx) + I|

Giai he (3), (4) la nhan diTrfc , , I| = — [ s i n ( l n x ) - c o s ( l n x ) ] + C

I2 = —[sin(lnx) + cos(lnx)] + C

(4)

B a i 3. Tinh nguyen ham I = In(cosx)

COS^ X

dx

p a t

u = In(cosx) dv = 1 dx

cos x

, sinx , du = dx

COSX '••il' V = tanx

1 = ln(cosx).lanx + jlan^xdx = ln(cosx).lanx + j

A.X i.

\COS^ X - 1 dx

= ln(cosx).tanx + t a n x - x + C

ằ "Jiiit loiiii ^lii" Itliiia

Tinh I = |P(x).sinadx hoSc JP(x)cosaxdx Ta lya chpn mot Irong hai each sau:

Ccich I: S\i dung lich phan tifng phan. Ta IhiTc hicn theo cac bu'dc sau:

'du = P'(x)dx Bi/(tc 1 : D a t u = P(x)

dv = sinaxdx v = — c o s ax a

1

If; A,,::.

Btf(tc 2; I = — P ( x ) . c o s a x + — P'(x)cosaxdx a a •'

Btf(?c 3; Tie'p luc nhU' Iren, ta khijf di/(Jc da ihuTc ^ i - Cuch 2: SOr dung phUclng phap he so bat djnh. Ta ihifc hicn theo cac h\i^^ ^'^^

Bif<tc 1 ; Ta c6: I = P(x)cosaxdx = A ( x ) s i n a x + B(x).cosax + C (1) Trong do A(x). B(x) la da ihtfc cung bac voti P(x). - Btfitc 2; Lay dao ham hai ve cua (1), ta dU"dc

P(x)cosax = (A'(x) - aB(x)]sinax + [ a A ( x ) + B'(x)]cosax Siif dung phU'dng phap he so bat djnh la xac dinh cac da thuTc A ( x )

\\\i(ic 3; Ket luan '1''' Nhan xel: N c u bac cua da thiJc P(x) Idn h(tn hoac bhng 3, la lha'y ng'^y '^'^f^

1 16 ra qua cong kenh, vi khi do, ta IhuTc hicn lay lich phan tifng phil'' "^ieu hrtn ba Ian. Do do, la di U'ii nhan dinh chung sau:

Neu bac cua P(x) nho htJn hoac bang 2, la lifa chon each 1.

Ncu bac cua P(x) Idn hiJn 2, ta liTa chon each 2.

Bai 4. Tinh nguyen hiim 1 = Jx.sin^ xdx

Hi/(}ng dSn giai 1 I = - f x ( l - c o s 2 x ) d x = -

2 •' 2 I I = Jxcos2xdx

xdx — 2-'

1 2 I , x.cos2xdx = —X —1 1

4 2

Dat u = X

dv = cos2xdx

du - dx

1 . - ''''yl V ==-sm2x :t r

2

, f , !i

L,uyfn giai ae iruac Ky ini vn j mien oac, i rung, i\am i oan npc - mguyen Van Thdng

=> I , = — x s i n 2 x -

=> I = — x s i n 2 x — c o s 2 x + C 4 4 8

sin2xdx = — xsin2x + — cos2x + C ,

2 2 4 '

1

0 + y - /, nr.) f f.iMil.{Km-y>i\'.

va

1 x ^ - 1 4 4

B a i S.Tinh I = JCx"* - x^ + 2 x- 3) . s i n x d x Ta c6: J ( x ' ' - x^ + 2 x- 3) s i n x d x

,.T ,\.ô , , - • , ' • • •

= (a|X"' +b|X^ + C | X + d|)cosx + (a2X-' + b 2 X ^ + C j X + d j )sin x + C (1) Lay dao ham hai v c c u a (1), la diTcJc: , , . , , , ^ , , (x'' - X ' + 2x - 3)sinx = [a2X^ + (3ai + b2)x^ + (2b| + C2)x + C| + djjcosx - [aix'' - (3a2 - b|)x' + (C| - 2b2)x - C2 + d|]sinx

[ a 2= 0 3a| + bj = 0 2b, + C | = 0 C| + d 2= 0 32 = 0; b2 = 3; C2 = - 2 ; d 2 = - 4

a, = - 1 ; b| = 1; C| = 4; d| = 1

K h i do I = (-x' + x^ + 4x + 1 )cosx + (3x^ - 2x - 4)sinx + C ftniii dim ldi6u: furj jurtj jiU ijj Tinh I = I c " ' cos(bx)dx hotic Je"" sin(bx)dx vcJi a, b ^ 0 Ta lifa chon mot Irong hai each sau:

- Cdch I: (Sur dung lich phan luTng phan) Ta IhiTc hicn ihco cac biTdc sau:

B i f i k 1;

du = - b s i n ( b x ) Dong nhaTl d^ng thiJc, ta du'dc

- a, = l 3a2 -b| = - 1

2 b 2 - C | =2

C2 - d , = - 3

.n

D a i u = cos(bx)

dv = c ' " d x v = i e "

a

K h i d 6 I = - c " cos(bx) + - f c " sin(bx)dx 1 _ r „ b r

a a I " ' (1)

Bif(?c 2; X c l I | = fc''\sin(bx)dx

D a i • u = sin(bx) d v - c ^ ' d x

du = bcos(bx)dx £ 1

xfj-

V = - c a

,ax

K h i d 6 : I , = i e ' " ' s i n ( b x ) - - f e ' " ' c o s ( b x ) d x = 4 " . s i n ( b x ) - - I (2) '"-^

a a a a Btf(?c 3; Thay (2) v^o (1), ta difdc

j ^ i c " " cos(bx) + -

a ^ i c " " s i n ( b x ) - - I

a a

_ [acos(bx) + bsin(bx)]e'''

a'+b^ + C

Cdch 2: Suf d u n g phiTcIng p h a p hp so bat d j n h , la ihiTc h i ^ n i h e o cdc biTdc sau:

I t i f ^ T a c o

1 = | c' " . c o s ( b x ) d x = [ A c o s( b x ) + Bsin(bx)]c'"' + C (3) i r o n g do A, B \k cdc hang so

Bif(tc 2: Lay dao ham hai ve cua (3), la du'dc • S

^li^^i^^x) = b l - A s i n ( b x ) + Bcos(bx)|c" + a|Acos(bx) + B s i n ( b x ) ] e "

= ((Aa + Bb)cos(bx) + (Ba - Ab)sin(bx)le

Aa + B b - 1 — Dong nha't dang thiJc, ta di/rtc

acos{bx) + bsin(bx)

B a - A b = 0 B -

a + b

Vay I = + C

Chii y: N c u bai loan y c u cau tinh gia trj cua mot cSp lich phan , . ' Ii = J c " c o s ( b x ) d x va I2 = Jc''\sin(bx)dx i h i ta ndn chpn cdch irinh bay

nhiTsau: '-'.

Suf dung lich phan liTng phan cho I| nhif sau: • ( t ^ ' Dai u = cos(bx)

d v ^ r e ^ ' d x

du = - b s i n ( b x )

a

1 K 1 K Khi do: I , = - e ' " ' c o s ( b x ) + - [ 0 " " sin(bx)dx = - e " " cos(bx) + - l 2 (3)

a a a a Sur dung lich phan tifng phan cho I2 nhiT sau:

du = bcos(bx) Dat u = sin(bx) 1

dv = e'"'dx V = - e a

Khi do = U - i c ' " ' s i n ( b x ) - - fc'"'cos(bx)dx = V ' ' s i n ( b x ) - - I , (4) a a •' a a TCr he phiTdng irinh (3) va (4), la di/pc

acos(bx) + bsin(bx)

Luyfn gUU di truOc thi DH 3 miin Bdc. Trung. Nam ToOn hoc - Ngt^n Van ThOng

2. PhUdng phap tren cung dugfc ap dung cho cac tich phan

I| = J c " s i n - x d x . I2 = Jệcos^xdx ' U B a i 6 . T i n h l i c h p h a n b a : i d i n h 1= fc^'Vcos^ xdx ? ? n ! > U n ' ; , r, i j ?

^ , Hif(}nK d a n (>iai . . , _ ,, ^

T a c o : I = ^ j c ' ^ d + cos2x)dx = ^ J c ' M x + ^ Je" cos2xdx = A;" + ^ I , (1) Cddi I.

- fc^Mx + - fe" cos2xdx = ^ c " + - Ii = Jc''cos2xdx

u = cos2x fdu = - 2 s i n 2 x d x , , ,

I , =e cos2x + 2 e sin2xdx (2)

X . I . . . X ' J

Dai

v = c"

dv = c dx

I2 = Jc''.sin2xdx J fu = sjn2x rdu = 2cos2xdx

•A

dv = e dx I2 = c \ s i n 2 x - 2

V = 0 "

c''cos2xdx = c ' ' 2 x - 2 I Thay ( 3 ) vao (2) ta di/dc a nu,< I'U l u i l l lit

1

(3)

I , = c ' ' c o s 2 x + 2 ( e ' ' s i n 2 x - 2 1 | ) o l , = - ( c o s 2 x + 2sin2x)c'' + C (4) 5

Thay (4) vio (1), ta difdc

1 = c" + - ( c o s 2 x + 2sin2x)c'' + € = - 0 " + — ( c o s 2 x + 2sin2x)o'' + C 2L 5 . 10

c*(l + cos2x)dx = (a + bcos2x + csin2x)e'' + C Lay dao ham hai vc cua (5), ta diTdc

(5)

- c ^ C l + cos2x) = (-2bsin(2x) + 2ccos2x)c'' + (a + bcos2x + csin2x)e''

2 nimtj^

' I J

a + (2c + b) cos 2x + (c - 2b) sin 2x c

Dong nhat ddng thtfc, ta diT^c

2a = 1

2(2c + b) = l o 2 ( c - 2 b ) = 0

1

b = 1 10 c = — 1

5 I = |^(5 + cos2x + 2sin2x)e'' +C

ằ 73tii Itn'f ttlii'u: T i n h I = Jp(x)c''dx Ta li/a '-'h*.'" TiOl irong hai each sau:

Stf dung lich phan liTng phan. Ta ihifc hi0n thco cac bifdc sau:

fdu = P'(x)dx -r

Dai

u = P(x) d v ^ c ' M x

1

a

Khi do: 1 = - P ( x ) c " — a fP'(x)dx

a i .10

Ticp luc nhu'trcn, la khiif diTdc da thrfc

Sijr dung phi/itng phap he so ba't djnh. Ta thifc hicn thco c^c birde:

Ki/<?c 1; Ta CO I = JP(x)e'"'dx = A ( x ) c ' " ' + C (1) Irong do A ( x ) la da ihi?c eung bac vdti P(x)

Bi/<tc 2; Lay dao h i m hai vc cua (1) ta difdc: PCxH;"" =[A'(x)+aA(x)]c'"' (2) Suf dung phU'dng phap hC' so ba't djnh diTdc A ( x )

Bi/ik- 3; K c l luan

Nhain xct: N c u bac cua da thtfc P(x) Wn hdn hoac bang 3, la lha'y ngay each 1 to ra qua eong kcnh, v i khi do ta thifc hiOn la'y tich phan nhicu hdn ba Ian.

Do do, di t(^i nhan djnh chung nhiTsau: ? tv / ^ . i i - Ncu bac cua P(x) nho hdn hoac b^ng 2, lifa chon c i c h 1. ..,,.

- Ncu bac cua P(x) Idn hOn 2, ta lira chon each 2. . , B a i 7 . T i n h

ạ I = j x c ' ^ d x b. J(2x-^ + Sx^ - 2x + 4)ế'dx

H i ^ n g d S n g i a i i; 1 0:

du = dx ; , a. Dat u = x 1

v ^ V "

[dv = c-^'dx 3

K h i d 6 I = i x e ^ " - ! P''dx = l x . c ^ ' ' - ! - e ^ ' ' + C

3 3 J 3 9

b. Ta co: I = J(2x-^ + 5x^ - 2x + 4)dx = (ax"* + bx^ + cx + d j e ^ " + C (1) Lay dao ham hai vc ciia ( i ) , ta duwc

(2x' + 5 x ' - 2x + 4)c^* = [2ax' + (3a + 2b)x^ + (2b 2c)x + c + 2d]e^*

,B6ng nha't ddng thtfc, ta diftJc:

2a = 2 3a + 2b = 5 2b + 2c = - 2 ' c + 2d = 4

|Chi d 6 I = ( x ' + x ^ - 2 x + 3)e'" + C

a = l b = l c = - 2 d = 3

Luy(n gidi di truOc thi DH 3 miin Bdc, Trung. Nam ToOn hoc • Nguyfn Van Thdng

"Bai todn eiiiti Unfa: Tinh I = J x " In xdx ; a - 1 Hitting dSn giai

Ju = Inx dv = x " d x

, a + l

du =—dx

X

a + 1 J

a + l X r X X

K h i d o : I - I n x - dx = l n x + a + l •'a + l a + l B a i 8 . T i n h I = J x 2 l n 2 x d x

Hif(}ng dSn giai

.(i+i V .'J

( a + l ) ' + C

u = ln2x dv = x^dx

d u - dx

3

'.A t>J , n ' J l J \ / d f l JUJ q j

il;gfii!

K h i d o - x - ^ ' l n Z x — fx^dx = - x ' 'l n 2 x— x ^ + C

3 3 J 3 9 ! ; 4. Nguyen ham c i c ham so hifu ti :um Hjrtrt j^fwrt,/ rinib n i

De xac dinh nguycn ham cac ham so hffu l i , ta can linh hoat lifa chpn mot irong cdc phifdng phap cd ban sau:

4.1. Phuang phdp tam thiic 6(2c hai

* Hai todn eliia Idida: Xac dinh nguycn ham cac ham so hffu t i dffa vao tam Ihffc bac hai.

Tren cd sd dffa tam thffc bac hai ve dang chinh t^c va dung cong ihffc sau:

+ C , a ; t O

a. xdx 1 ,

= — I n

x ' ± a 2 x^ ± a + C Bai 1. Tinh tich phan I = xdx

. f dx 1 , X - a x + a

x ' ' - 2 x 2 - 2

Hff(Jng d i n giai xdx 1 f d ( x ^ - l ) x ' ' - x^ - 2 ^ x ' - i r - 3 2- ' ( x '- 1 ) ^ - 3

4V3 in - 1- V^ I

X ^ - I + N/3 + C Bai 2. Tinh tich phan I - J — x^dx

x ' ' - x ' - 2 3

COng ty TNHH MTVDWH Khang Vu Hiiidng d i n giai

T a c 6 : I = j - x^dx 2 ; + -1

9 2

2)

x ^ - i

x=-'.^

2 ;

4 ' x ^ - i ^

\ I J:

9 2 ;

9 4 J /

4

2J + — I n 12

2) 3 2 " 2

9 i;.f|.t.

4 2 1 3

X

2 1 3

X - - + -

2 2 + C

^ J . Phuang phdp phdn ttch

CJ day, thffc chat no \k mot dang cua phffdng phdp he so ba't d|nh Oidi ftfdii ehiu Uuia: Tinh tich phan I - j —

Sff diing dong nha'l thffc

(ax + b)" -dx ; a ?t 0; a 0

2 1 2 2 1

X =—a X = —

3. H

Khi do:

(ax + b ) - b f - - ^ " ^

1 (ax + b r- 2 b( a x + b) + b^

(ax + b)^ - 2 b( a x + b) + b^

=>1 =

(ax + b)" a^ (ax + b)

1 1 ' 2b b^

a^ (ax + b r ^ (ax + b ) " - ' ' (ax + b)"

1 " c dx f 2bdx f b^dx a" •'(ax + b ) " ' ^ •'(ax + b ) " " ' ' -"(ax + b)"

[• d(ax + b) |-2bd(ax + b) [•b^d(ax + b) J(ax + b)"-2 ' ( a x + b ) " - ' (ax + b ) "

Bai 3: Tinh tich phan I = f— j ^ r d x ,

• " ( I - x ) ^ ^ • n i : Hif(}ng dSn giai

Sff dung dong nhal thffc x ' = (1 - x ) ' - 2( 1 - x) + 1 x ' 1 2 1

• + •

( 1- x ) " ' ( 1- x ) " ( i - x ) - ' " ( 13X - x )

Luyfn gUu di truOc thi DH 3 miin Bdc. Trung. Nam ToOn hpc - NguySn Vdn Thane

Khi do I = dx t 2dx f dx

" J , , ^ ill - v ^ y

( l - x ) " M - x r M - x ) 1

3 6 ( 1 - x ) ' " 3 7 ( 1 - X ) " 3 8 ( 1 - X ) .17 B a i 4. T i n h lich phan I = j—^ ^ ^ d x

( x - l ) '

HxiHn^ dan Kiai

Suf diing dong nhal ihiJc x ' = 1 + 3(x - 1) + 3(x - I + (x - 1)' T a d m - : _ l ^ 3 ( x - l ) . 3 ( x - l ) ^ . ( x - l ) ^

( x - l ) Id ( x - l ) II)

I 3 1

( x - l ) ' " (x-1)'^ ' ( x - l ) * * ' ( x - 7 ) ^

1 3 3 1 dx

• + - + - +

( x - l ) ' " ( x - l ) ' ' ( x - l ) * * ( x - 1 ) ^

1 3 3 1

9 ( x - l ) ' ^ 8 ( x - l ) ' * 7 ( x - l ) ^ 6 ( x - l ) ' " + C 4.3. Sa dungphuongphdp tkh phdn tCtngphdn

Phifi/ng p h a p : N c u lich phan can l i m c6 dang I = jP^^'^Q'^") Ta thifc hicn theo cac bifclc sau:

u = P(x)

Q"(x) dx

B t f < t c l ; D a t

dv = -2 dx Q"(x) Bif<k 2; K h i do 1 = uv - j v d u

du V

B a i toan 5. Tinh tich phan I = f — — ^

• ' ( x - - l ) ^

Hxi^n^ dSn giai Bic'n ddi I , v6 dang I = f - ^ ^

^ x - -1)^

D a l

II = X

dv = xdx =>^

( x ^ - I ) ^

du = 3 x ' d x V = - 1

4 ( x ^ - l ) ^ K h i do I = + l f - ^ ' ^ ^

,2 _ n A J,

4 ( x ' - l ) 4 - ' ( x ^ - l ) 2 (1)

x ' d x 1 f[(x + l ) + ( x - l ) f

4 J r v -

( x ' - l ) 1

dx

( x - 1 ) ^ x ' - l ( l + x ) " J dx 1

X - l • + ln x - l T i r ( l ) . (2) suy ra 1= -4 ( x 2 - l ) ' 16

x + 1

In 1

X + 1 ^ + C = -

4 In X - l X + 1

2x

x - l j + C (2)

x - l x + 1

2x ^ x ^ - 1 + C

yV/jfl/i x^/: Dc xiic dinh lich phan l i , la ciing c6 ihc suf dung tich phan lu'ng phan nhiTsau:

Dai <

u = x dv = xdx

( x ^ - 1 ) ^

du = dx ••nl

V = — 1

Khi do: I ,

2 ( x ' - l )

dx X

1 f u x X + - l n 2 ( x ^ - l ) 2 - ' x ' - l 2 ( x ' - l ) 4

x - l x + 1

5. Nguyen h a m c a c h a m so li/Ong giac i Dc xac djnh nguycn ham cac ham so liTclng giac, la can linh hoal Ufa chon

mol Irong cac phu'dng phap c(J ban sau:

5.1. Sii dung cdc phep biSn ddi lupng gidc dua vi cdc nguySn hdm ca ban.

f dx

* Hai ioun t-liiu tditUi: T i n h lich phan 1 = sin(x + a)sin(x + b) Ta Ihifc hicn cac biTdc sau:

, s i n ( a - b ) sin[(x + a ) - ( x + b)]

J dong nhal Ihifc 1 = = — ' — : • Btfrifc 1; Siif dung ddng nhii't ihuTc

Btf(tc 2: Ta difdc

I |-sin[(x + a ) - ( x + b) s i n ( a - b ) •'sin(x + a).si

sin(a - b) sin(u - b )

1 = dx

1

sin(x + b)

sin(x + a).cos(x + b ) - c o s ( x + a)sin(x + b) sin(a - b) •' sin(x + a).sin(x + b)

1 rcos(x + b) , ccos(x + a) s i n ( a - b ) ^ sin(x + b)

I + fCos(,x + i dx -

.+ b) -"sinCx + a)

-dx = 1

sin(a - b) •In sin(x + b) sin(x + a) + C Nhqn xet: Phiftlng phap ircn cung diTdc ap dung cho cac dang lich phan sau: