cẩm nang luyện thi đại học nguyên hàm tích phân số phức

LOI N O I £>AU I Tap sdch nay gom 2 phan: Phan I: Nguyen ham-Tich phan vaung dung Phan 11: So phuc Moi phan diroc trinh bay theo tung chirong, moi chuong bao gom cac chuyen de, moi

C-UON T R A N B A H A

AM NANG LUYI^THI DAI HOC IIGUYEN HOM

* * THU VIENTINHBINHTHUAfO

mk nXr ikn i«i n c wfc cu M Mi

LOI N O I £>AU I

Tap sdch nay gom 2 phan:

Phan I: Nguyen ham-Tich phan vaung dung

Phan 11: So phuc

Moi phan diroc trinh bay theo tung chirong, moi chuong bao gom cac

chuyen de, moi chuyen de du-oc phan thanh cac van de co ban, moi van de

bao gom: Tom t5t kien thurc - phuong phap giai - bai tap ap dung - bai tap

tu luyen Cuoi moi chvrong deu co phan Bai tap tong hop va Bai tap luyen

thi bao gom cac bai tap nang cao duoc tuyen chpn qua cac de thi dai hoc va

cac de thi hoc sinh gioi

Hi vong rang tap sach nay co the giup ich cho hoc sinh trong cac ki thi

hoc sinh gioi, ki thi dai hoc Rat mong su gop y cua doc gia va dong nghiep

de Ian xua't ban sau tot hon

Trdn Bd Ha Gido vien THPT Chuyen Le Quy Don - Dd Ndng

Tu nghiep tgi: lustitut de Recherche Pour L 'enseignement des Mathe 'matiques

Paris-France

Nha sach Khang Viet xin tran trong giai thieu tai Quy doc gia va xin

long nghe moi y kien dong gop, decuon sdch ngay cang hay han, botch han

Thuxinguive:

Cty T N H H Mpt Thanh Vien - Dich vu Van hoa Khang Vi?t

71, Dinh Tien Hoang, P Dakao, Quan 1, TP H C M

Tel: (08) 39115694 - 39111969 - 39111968 - 39105797 - Fax: (08) 39110880

Hoac Email: khangvietbookstore@yahoo.com.vn

Cty TNHH MTV D VVH Khang Vi^t

2 Tinh chat co ban:

+ Neu F(x) la mot nguyen ham cua f(x) tren D thi F(x) + C cixng la nguyen ham ciia f(x) tren D (C la hang so)

+ Neu F(x) va G(x) la cac nguyen ham cua ham so f(x) tren D thi ton tai hSng soCdeG(x) = F(x) + C

+ Ky hieu: jf(x)dx = F(x) + C(laho nguyen ham ciia ham so f(x)) + Neu f(x) va g(x) co nguyen ham tren D thi:

l[f(x) + g(x)dx = Jf(x)dx + l(x)dx

jkf(x)dx = kjf(x)dx, ke R

+ Neu Jf(x)dx = F(x) + C thi Jf(ax + b)dx = - F(ax + b) + C

a + Moi ham so lien tuc tren D deu co nguyen ham trenD Y<.'

3 Bang cong thiic nguyen ham ca ban:

C ldx = x + C f w ^ ^

je'""dx= + C n ; Jx«dx= + C a ^ - 1 )

C 'dm nang luy^n thi DH - Nguyen ham - Tich phdn - So phm I/uui Bd Ha

B C A C D A N G T O A N C O BAN:

Chuyen del: K H A I N I E M N G U Y E N H A M

Van de 1: C H l / N G MINH F(x) LA MQT NGUYEN HAM CUA f(x) TREN D

Phuang phap: Chung minh F '(x) = f(x), Vx e D

B a i l : Chung minh: F(x) = ln(x + V x " +1) la mot nguyen ham cua:

Vay F(x) = ln(x +Vx^ + 1 ) la mot nguyen ham cua f(x) tren R

Bai 2: Chung minh F{x) = xsinx + cosx la mpt nguyen ham ciia:

f(x) = xcosx tren R

Giai

F '(x) = sinx + xcosx - sinx = xcosx Vx e R

Vay: F(x) = xsinx + cosx la mot nguyen ham cua f(x) = xcosx

- 2 1 Vay F(x) = —j= la mpt nguyen ham ciia f(x) = — j = r

Vx xVx

Bai 4: Chung minh ham so F(x) = 1 1

3cos'x cosx la mpt nguyen ham ciia

ham so: f(x) = ^ ' " ^ tren mien D = R \ - + kn; k e Z)

cos X 2

Vx 7t — + kn, ta c6:

2

Ljy li\tin Ml V uv vn t^^nungnci

sinx sinx s i n x ( l - c o s " x) sin^ x

F'(x)= — 1 = = — r - = t ( x )

cos X cos X cos X cos X

Vay F(x) la mpt nguyen ham ciia f(x) •

2 + x Bai 5: Chung minh F(x) = x In + 21n(4 - x^) la mot nguyen ham cua ham so

f ( x ) = l n ^ ^ tren(-2;2) v-^

2 - x

Giai , 2 + x 4x 4x , 2 + x , ^ , ^

Taco: F'(x) = i n - + - 5 - - - ^ = l n - - t ( x ) , V x 6 ( - 2 ; 2 )

2 - x 4 - x 4x- 2 - x

Do do: F(x) la mot nguyen ham ciia f(x)

asinx + bcosx Bai 6: Chung mmh ham so f(x) = (c^ + > 0) co nguyen ham

csmx + dcosx dang: F(x) = Ax + Bin I csinx + dcosx | + C

Giai ccosx - d s i n x (Ac - Bd)sinx + ( A d + Bc)cosx

Ta c6: F '(x) = A + B = -^^ ^

csmx + dcosx csmx + dcosx F(x) la nguyen ham cua f(x) <=> F '(x) = f(x), Vx

o (Ac - Bd)sinx + (Ad + Bc)cosx = asinx + bcosx, Vx {A c - B d = a ^ , ac + bd „ b c - a d

Giai he ta co: A = — — - y ; B = — —

A d + Bc = b c - + d ^ c ^ + d ' Vay ho nguyen ham ciia f(x) la:

ac + bd be - ad , ^ ^

F(x) = r - + — r- In csinx + dcosx + C

^ ' c ' + d - c - + d ' Bai 7:

a) Tim a, b, c sao cho ham so: F(x) = (ax^ + bx + c) V2x - 3 la mpt nguyen ham ^ ^ 2 0 x - - 3 0 x + 7 ^ 3

cua ham so: f(x) = , tren ( - , « )

V 2 X - 3 2 b) Tim nguyen ham G(x) cua f(x) thoa man man G(2) = 0

Giii

ax^ + bx + c a) Ta co: F '(x) = (2ax + b) V 2 x - 3 + ,

V 2 x - 3 (2ax + b)(2x - 3) + ax- + bx + c Sax' + (3b - 6a)x + c - 3b

V 2 X - 3 ~ V 2 X - 3

5

ihiiiy, liivci] nil - Xyjiv'ii lu'iiii - Tirh f)ficiii - Soptiijc - Trcin Bci Ha

F(x) la nguyen ham ciia f(x) <=> F '(x) = f(x), Vx e (—; oo)

Ta c6: F '(x) = (2ax + b)e'^ + (ax^ + bx + c)e'' = (ax^ + (2a + b)x + b + c ) ^

De F(x) la nguyen ham ciia f(x), Vx e R thi phai c6

Bai 9: Tinh dao ham ciia ham so': F(x) = (x^ - 1) In 1 + x - x^In

T u do suy ra nguyen ham ciia ham so': f(x) = xln

Xet G(x) = F(x) + X => G '(x) = F '(x) + 1 = f(x) nen G(x) la nguyen ham ciia f(x;

Vay: G(x) = (x^ - l ) l n I x + x | - x^ln I x I + x la mot nguyen ham ciia f(x)

Bai 10: Chung minh F(x) =

la nguyen ham ciia f(x) =

k h i x ; t 0 khi X = 0 ( x - l ) e ^ + 1

K ^ u i i i iiuii^ luy^n ini uii - ivguyeri nam - i icnpmm - AOpmtC - Iran tsa na

Vamde2:TIM HQ NGUYEN HAM CUA HAM SO y = f(x) BANG D I N H NGHIA

Phuong phap: Phan rich f(x) thanh tong (hieu) cua cac ham so' ca ban c6 the

tim nguyen ham bang each ap dung bang cong thuc nguyen ham ca ban, ap

dung rinh cha't cua nguyen ham de tinh hoac dua ve dang nguyen ham cua

ham so'hgp

Bai 1: Tim hp nguyen ham cua f(x) = cosxcos3x

Giai f(x) = ^ [cos4x + cos2x]

Jf(x)dx = — J(cos4x + cos2x)dx = — [ — sin4x + — sin2x] + C

a) (sinx + cosx)^ b) sin-'x + cos^x

c) sin''2x + cos*2x d) cos'^x

= l((sin22x + cos22x)2- 3sin22xcos22x) = 1 - - sin24x

Bai 6: T i m hp nguyen ham ciia cac ham so':

a) cosx.cos2x.sin4x b) cos'x.sinSx

Giai a) Ta c6: cosxcos2xsin4x = ^ [cos3x + cosx]sin4x Ai»

= — [sin4x cos3x + sin4x.cosx] = - [sin7x + sinx + sinSx + sin3x]

Do do: fees X cos 2x sin 4xdx = — f(sin 7x + sin 5x + sin 3x + sin x)dx

J 4 J

+ C

1 + cos2x^

4 ' —cos7x+—cos5x+ —cos3x+ C O S X

7 5 3

4^

b) Ta c6: sinSx.cos-'x = sinSx cosx

= ^ [sinSx.cosx + sin8xcosx.cos2x] = ^ sinSxcosx + ^ sin8x(cos3x + cosx)

= — (sin9x + sin7x) + — sin8xcos3x + — sinSx.cosx

88 24 56 49 Bai 7:

3x +1 A B a) Xac djnh cac he so' A, B sao cho: ^ = ^ +

(x + 1)' (x + 1)' (x + 1)'

3x + 1

b) Suy ra ho nguyen ham ctia ham s6'f(x) = —'-

r-(x + 1) Giai

= 64

Giai ''sin" 2x + cos^ 2x^

sin"2x = 64

cot^2x^ , sin 2x sin 2x , / 1 cot" 2x cot" 2x cot^ 2x

sin 2x sin 2x sin 2x snr 2x

2 4cot-2x 2cot^2x

3M sin"2x sin"2x sin"2x

Do do: I = 32Icot2x + - cot^2x + - cot^2x] + C

3 5 Bai 9: Tinh

a) T i m nguyen ham F(x) cua fn(x) thoa man man F(0) = 1 Suy ra bieu thuc thu gon

cua fn(x)

b) C h u n g m i n h : gn(x) = f 'n*i(x) T i m bieu thuc thu gon ciia gr.(x)

G i a i a) fn(x) = 1 + 2x + 3x2 + + n X " - ' F(x) = x + x^ + x^ + + x" + C

n(n + l)

khi X ^ 1 khi X = 1

b) fn+i(x) = 1 + 2.x + 3x2 + + n.2"-' + (n + l ) x " suy ra:

4 4 Hay C = 1 + ; Vay: F(x) = -cotx - x + 1 + ^

Bai 2: Cho f(x) = sin^x(1 + tanx) + c o s \ ( l + cotx)

T i m nguyen h a m F(x) cua f(x) biet F( — ) = 1

4

G i a i Rut gon f(x) ta c6: f(x) = sinx + cosx Jf(x)dx = sinx - cosx + C => F(x) = sinx - cosx + C

F ( - ) = l < : ^ s i n - - c o s - + C = 1 « C = 1

4 4 4 Vay F(x) = sinx - cosx + 1 Bai 3: Cho f(x) = ? T i m nguyen ham F(x) biet F( - ) = 0

l + cos2x 3

G i a i f(x) = = —

Cam iwn}^ luy('n thi DH - Nguyen ham - Ticli plum - So phi'rc - Trdn Bd Ha

Bai 4: C h o f(x) = ^ ^, F(x) la mot nguyc"n ham cua f(x) thoa man m a n : F(2) =

a) C h i i n g m i n h F(x) = tanx In(sinx) - x la mot nguyen h a m cua:

f(x) = (1 + tan^x) In(sinx) tren (0; ^ )

b) T i m nguyen ham F(x) cua f(x) bie't F( —) = —

D o d o F(x) = tanx In(sinx) + ^ Xnyjl , *

Bai 7' C h o f(x) = — \ — T i m nguyen ham F(x) ciia f(x) bie't d o thj ham so:

X - 1 Bai 8: Cho bie't F(x) = la nguyen ham cua f(x) T u n f(x - 1)

x + 1

f(x) = F ( x ) =

G i a i

^ x - l V 2 , x + 1 ( x + 1)-^

2

D o d o : f ( x - l ) = — r

Cdm nang luyeti ihi DH - Nguyen ham - Tich phciii - So phi'rc - Trdn Ba Ha

Chuyendel: P H l / O N G P H A P T I M N G U Y E N H A M

Van de 1: T I M H Q N G U Y E N H A M B A N G P H U O N G P H A P D O I B I E N S O

Phuong phap: Co the doi bien so theo hai each sau:

a) Dat u = cp(x) la ham so'co dao ham thi:

Jf (x)dx = |f(u)du =F(u) + C

b) Neu f(x) Hen tuc c6 dao ham, dat x = (p(t) th'i: Jf(x)dx = Jf{(p(t))(p'(t)dt

B a i l : Tim ho nguyen ham cua cac ham scYsau:

cosx Giai

Ccim nang liiyen thi DH - Nguyen ham - Tich phdn - So phiic - Trdn Bd Hd

Giai a) Dat u = In I In(lnx) | du = i l l l i l E ^ j x

V a ' - x" = aVcos" t = a(cot) = acost

fVa' - x" , f a cos t.a cos tdl rcos" t ,

( iiiir.yi /in cii llii /'// - A i v / i iv; • ;/ - / A /; plum - So phllV BaHa

V a n de 2: T I M H Q N G U Y E N H A M B A N G P H l J O N G P H A P N G U Y E N

H A M T l / N G P H A N

Phucmg phap: Gia six u(x), v(x) la cac ham so c6 dao ham lien tuc khi do ta c6:

Ju(x)v'(x)dx = u(x)v(x) - v(x)u'(x)dx

hay udv = uv - vdu

Chu y: Cac dang sau:

+ P(x) Inxdx: Dat u = Inx, dv = P(x)dx

+ P(x) sin(ax)dx , P(x) cos(ax)dx: Dat u = P(x), dv = sinax (cosax) dx

+ |P(x) e-^^dx: Dat u = P(x), dv = e-'^dx

+ e"" sin(bx)dx hoac je'" cos(bx)dx v

Dat u = e-'"; dv = sin(bx)dx hoac dv = cos(bx)dx

+ Tong quat: Phan tich f(x)dx thanh u va dv sao cho: tu dv suy ra duq^c v va

Jvdu don gian hon

Bai 1: Tim hp nguyen ham ciia cac ham so'sau:

Do do: J = - (x + 1 )2sin2x + |(x + l)sin 2xdx

Xet A: (x + l)sin2xdx Dat u = x + 1 =>du = dx

dv = sinZxdx => V = — cos2x

2

A = - - i ( x + l ) c o s 2 x + - fcos2xdx = - - ( x + l)cos2x + ^sin2x + C

Cam nang luyen thi DH - Nfjiiyen ham - Tich phiin - Sd phirv - Trdn Bd Ha

Vay |f(x)dx =

6 2 ^ ( x +1)^ sin2x - ^ ( x + l)cos2x + l s i n 2 x + C

^ O i : ^ 2 L + I ( x + i)2 s i n 2 x - i ( x + l)cos2x + i s i n 2 x + C

6 4 4 8 Bai 3: T i m ho nguyen ham cua cac ham so'sau:

dv = sin xdx => v = -cos xdx do do I = -o^cosx + j e " cos xdx (1)

C / y IWlltl mi V UV Vll l\.nun^ r tcr

XetJ= j e " C O S x d x Dat u = e" => du = e"dx

dv = cos xdx => V = sin xdx e"" sin xdx = e^sinx - 1

Giai < : - ci

dv = sin^ X - V = - c o t x

a) f(x) = e"2^cos3x b) f(x) = sin(lnx)

Giai a) Dat u = e-2>= d u = -2e-2^dx - ifs.:c 1 > ' ?'/ ^ /K^*

Vay A = ~" (xsin(lnx) - cos(lnx)) + C

Bai 7: T i m ho nguyen ham ciia cac ham so:

180 81 180 81

25

Ccim ncing hiyen ihi DH - N}^iiycn ham - Tich plidn - So phirc Trdn Ba Ha

Chuyen de3: N G U Y E N H A M CUA C A C H A M S O CCf B A N

Van de 1: N G U Y E N H A M C U A C A C H A M SO H U U T I

P(x) Phuang phap: De t'lm ho nguyen ham ciia cac ham so'dang , voi P(x),

Q(^) Q(x) la cac da thiic ta thyc hien: , i ; 5.^ I T ?

+ Ne'u bac ciia P(x) nho han bac cua Q(x) th'i phan h'ch Q(x) thanh tich cac

thira so bac nhat va bac hai roi phan tich thanh tong (hieu) ciia cac phan

thuc don gian (mau so' la cac thiia so' bac nhat, bac hai 6 tren) de tim ho

nguyen ham

+ Ne'u bac cua P(x) Ion hon hoac bang bac cua Q(x) thi dung phep chia da

thuc de du'a ve truong hop tren

+ Chu y cac ho nguyen ham C O ban '>

CIV TNIIIIMTVDVVHKhang Viet

gai 2: T i ' " ho nguyen ham cua cac ham so'

Bai 1: Tim hp nguyen ham cua

Khix = l = 5 B = - 4 ; x = 2=>C = 7

x = - l => A = -7

( x - i r ( x - 2 )

Vay 3x + l 4 dx = - 7 1 n x - l + + 7 1 n x - 2 + C ( x - l ) - ( x - 2 ) x - l

Cam nang luy^n thi DH - Nguyen ham - Tich phdii - So pht'rc - Trdn Bd Ha

Do do: k = x + 1 dx X + 1

2 ( 2 x - l ) ' 2 - ' ( 2 x - l ) - 2 ( 2 x - l ) - ' 4 ( 2 x - l ) + C

Bai 7: Tinh M =

x- + X + 1 Dat

X" + X +1 ( x - l ) -

cx + d

29

Cam nang hiyen /hi DH - Nguyen ham - Tich phcin - So phi'cc - Trdn Bd Ha

b) H p nguyen ham dang: J R X, Vax" + bx + c Jx

Neu a > 0: dat t + V a x = Vax" + bx + c

Neu c > 0 : dat xt ± - / c = V a x ' + bx + c

Neu ax^ + bx + c c6 nghiem xi, X2, dat Vax' + bx + c = t(x - xi)

(Truong hop a < 0 hoac c < 0 thi dat x = — dO diia ve dang tren)

u

Chu y: Co the h u u ti hoa bang each bie'n doi

roi dat II = X + -— de dua ve cac dang:

2a ax2 + bx + c = a X + — b A

J R , ( U , Va' - i r )du : cTat u = asint

c) H o nguyen ham dang: f (Ax + B)dx

t = - 3 ^ 1

= X => dx = — 2(t + l) 2

Cdm nang luy^n thi DH - Nguyen ham - Tich plidii - So phi'rc - Trdn Bd Ha

ix + 3 + -\/x- +6\ 8 E= f - ^ = lnlt + 3 + C = ln

Jt + 3 '

Bai 6: Tinh F = J V x " - 4 x + 8dx

+ C

Giai Dat t - X - A = - 4 x + 8 X = - ^ ^ - ^

b D^ng bac cao theo sinx, cosx, tanx: Bien doi ve d^ng chiira hai nhom ham Itr^ng giac c6 lien quan dao ham roi diing doi bien so'de tinh hoac dung cong thuc ha bac de dua ve nguyen ham cac ham lugng giac co ban

p ^ l ; Tim nguyen ham aia cac ham so sau:

I 1+sinx

a)f(x) =

cosx b ) f ( x ) = sin x (l+cosx)

Giai , f dx rcosxdx fcosxdx

i) I = I = I 5— = r - r

""cosx "'cos^x • ' l s i n D$t u = sinx ==> du = cosxdx, ta co: I = 6 : 1 - J ^

-2 - ' V l - u 1 + u ; du = —In

2

l + u 1-u + C = - l n 2

1

1 + sin x + cos X b ) f ( x ) = - r V -sin

X

Cdiii ncing luyen thi DH - Nguyen ham - Tich phuii - So phi'rc ~ Trdn Ba Ha

t-1 + - y + , 1+t- 1 + t-

Bai 3: Tim nguyen ham cua cac ham so sau:

cosx a) f(x) =

a) f(x) =

1 3 - l O s i n x - cos2x

cosx

COS" X b) f ( x ) = -

sinx Giai

cos X

13-lOsinx - ( 1 -2sin- x) 2sin- x -lOsinx + 12

Dat t = sinx => dt = cosxdx, do do:

ny_ TNHH MTV D VVH Khang Vi^

Suy ra: cos'x^ f ( l - t ' ) ' d t -dx =

sin X

1= flzZL±l!<it= ((i-2t + t'ldt = l n t - l ^ + - + C

I = In I sinx I - sin^x + — sin^x + C

Bai 4: Tim ho nguyen ham cua cac ham so:

a) f(x) = cos''xsin^x

Giai a) Jcos^ xsin' xdx = Jcos^ x(l - cos" x).sin xdx Dat t = cosx => dt =-sinxdx Do do:

b)Tac6: - 1 sin"x + cos-x I cot"\

sin"* X sin"* X sin^x sin'x - + •

Do do: cof^x = — + cot^ X , + cot' x — + — — +1

sin X sin" X sin x sin x

Cam nang lny^n thi DH - Nguyen ham - Tich p/ic'in - So phiic — Tran Bd Ha

a) f(x) = cos^x b) f(x) = cos*x

Giai a) Taco:

cos''x • ri + cos2xV ^_i_r,

4 1 + 2cos2x + COS" 2x l + 2cos2x +

l+cos4x

- [ - + 2cos2x + - cos4x]

4 2 2

Suy ra: Jcos' x dx = ^ [ ^ x + sin 2x + ^ sin4x] + C

cos* x d x Dat I u = cos' xdx => du = - 5 cos'' x sin xdx

i = cos'^xsinx + 5

dv = cos xdx => V = sin X

cos' X sin^xdx = cos^xsinx + 5 cos' x (1 - cos'x)dx

= cos'xsinx + 5 Jcos' xdx - 5 Jcos^' xdx

)iung luyirii - i^yny>in nuiii - i icn jnuni - M)pniK - JYdn lid tia ~

C B A I T A P T O N G H Q P VE N G U Y E N H A M

1 B A I T A P TV" L U A N

Bai 1: Cho f(x) = x V 3 - x Tim a, b, c de ham so F(x) = (ax^ + bx + c ) V 3 - X la

mot nguyen ham cua f(x)

Giai Taco: D y = (-co; 3]

ax^ +bx + c _ -5ax- h(12a - 3 b ) x + 6 b - c

F{x) la nguyen ham cua f{x) <=> F '(x) = f(x), V \ Dy

B a i 6; T i m ham so y = f(x) biet r3ng: f (x) = ifx + + 1 va f(l) = 2

Vay F(x) la mot nguyen ham ciia f(x)

Bai 8: T i m ham so'y = f(x) biet rang f '(x) = taii\.sin2x va f( —) = —

4 4

f '(x) = tanx.sin2x •• s m x

Gi.ii 2sinxcosx = 2sin x cosx

i ) f f ( x ) d x = - f ^ ( ^ ' " ^ + ^ " ^ ^ ) ^ _|„i^i,, ^ + ^.^^^X + C

J •' s i n x + c o s x

b) f<x)=cot- 2x +

-4 j sin"(2x + - )

sin' X

1 + sin X

Ctim naug /iivi'ii ihi DH - Nguyen ficiiii - 'llcli plum - So phi'rc - Trdn Bd Ha

J t ' ~ 4t^ + C = -4 1 n ' x + C

Bai 15: T i n h B = Jx 'Vl -x''dx

Giai Dat t = 1 - x'' => d t = -Sx^dx <=> x^dx = d t

Dat t = cosx ^ E = -1(1 - t2)f dt = 4 l ( - l ' + t'')til

E = - - t ^ + - P + C = - - cos'^x + - cus'x + C

Bai 19: Ti'nh I : Jsin x-Jlcos x - Idx

Gi.ii Dat t = 2cosx - 1 => d t = -2sinxdx

Bai 21: D u n g p h u o n g phap lay nguyCn honi lung phan hay t i m hp nguyen

ham ciia cac ham so':

Ta c6: [x'^e" - nin-i]' = n.x"-' e" + e^x" - n.x" ' = e« x"

Do do: x^e" - nln-i = Jcx'^dx = L (dpcni) b) t = l x V d x

11 = xe" - lo = xe" - e" = (x -1)6" + C

12 = x^e" - 2I1 = x^e" - 2(xe'' - e") + C

Cdni >icin^ luyen ihi DH - Nyiiyeii ham - lichphim - Sophi'rc — Tran Bd Ha

G i a i

X e t B =

cos X sin"* X + cos' X

Cam nang luy^n thi DH - Nguyen ham - Tic/i pluiii - So phuc - Trdn Ba Ha

C i a i

2 3

'-Vx e (-2; 2) ta c6: F ' ( x ) = - ( 4 - x " ) - 2.\

3 2 F'(x) = 2x V 4 - X - =f(x)

l - 2 s i n 2 x

<:> f(x) = sin2x + 2sin2xsinx = sin2x + C()sx - i.\is3x

|f(x)dx = - — cos2x + sinx - — sin3x + C F ( \

Bai 36: T i m mot nguyen ham cua ham s.V f(\ s i n x \ / c o s x biet nguyen ham

tri^t tieu khi x = 7t

Cam nang luyen thi DlI \'i;^iiyri; ham rich i>lh!ii So phin Iran Bd Ha

- Vl + x^ = f(x), Vx G R

Bai 38: Cho f(x) = sin3x(l + cotx) + cos''x(l + tanx) (0 < x < —)

Tim nguyen ham F(x) biet F( —) = 0

Giai

f(x) = sin"* x 1 + cosx

sinx + cos"^ x 1 +

sinx cosxy

= sin^x + cos^x + sin^xcosx + cos^xsinx

= sin^x(sinx + cosx) + cos2x(sinx + cosx)

Bai 41: Cho biet If(x)dx = In^x + C tim f(x):

Cam nang luy$n thi DH - Nguyen ham - Tich phdn - So phuc — Trdn Bd Ha

Bai 46: Cho f(x) = sin2x Tim nguyen ham F(x) biet F( —) = 0

F ( - ) = -ln>/2 +C = 0 « l n > ^ = C « C = - l n 2

F(x) = -In I sinx + cosx | ~ '"2

Bai 53: Cho Jf(x)dx = In ^ = + C tim f(x)

x + V x ' + l

Giai

ff (x)dx = In ] + C => F(x) = In ]

x + Vx^+1 x + Vx^+1 oF(x) = -ln(x+ Vx^+1)

(x + V x ' + l ) ' 1 f(x) = F'(x) =

x +Vx^ +1 Vx- +1 Bai54: Cho f(x) = r—?——. Tim nguyen ham ciia f(x)

Giai Jxcosxdx = Jxd(sinx) = xsinx - Jsinxdx = xsinx + cosx + C

MiSZ: Cho biet F(x) la nguyen ham ciia f(x) = — ^ va F(2) = 2 tim F(-2):

j j O : Cho biet F(x) la mpt nguyen ham ciia f(x) = tanx.sin2x thoa man

Giai

f(x) = sin''x.sin2x = 2sin'*x.cosx

2 ' If(x)dx = 2/sin''xd(sinx) = - sin'^x + C

Bai 62: Tinh

Giai xVx + ldx

54164: Cho ff(x)dx = - - I n coskx + C Tim f(x)

Shi rr" hp nguyen han, cOa f(x)

-Bai 7: T i m hp nguyen ham ciia f(x) = tan

g a i j : T i m hp nguyen ham ciia:

Ccim nang luyen thi DH - Nguyen ham - Tich phdn - So phirc — Trdn Ba Ha

1 sin X + C O S X + sin X - c o s x 1 sin x - c o s x

2(sin x + cos x )

+ C

c) F(x)= — s i n 4 x + — s i n 8 x + — + C

^ ^ ^ 32 128 16

Bai 3: Tinh F '(x) rut gon j V x " + l d x = ^ InVx" +1 + l n X + N/X^ +1

Bai 4: Dong nhat da thiic ta duoc A = 3, B = 2, C = l

Chvrong 2 : T I C H P H A N

A T O M T A T L Y T H U Y E T

1 Dinh nghia: Cho h a m so f lien tuc tren khoang I, a, b la hai so' ba't ky thuQC I

Ne'u F(x) la n g u y e n h a m cua f(x) thi hieu so: F(b) - F(a) du-gc ggi la tich

h

b

phan cua f(x) tir a den b va ky hieu: f (x)dx = F(x)

2 Dinh ly 1: Cho ham so y = f(x) lien tuc khong am tren khoang I va a, b la hai

so thuoc I (a < b) Dien tich hinh thang cong gioi han boi do thj y = f(x) tryc

b

Ox va hai duong thang x = a, x = b la: f (x)dx

a

3 Tinh chat cua tich phan:

Gia sir f(x) va g(x) lien tuc tren I va a, b, c la ba so'ba't ky thuoc I Khi do ta c6:

Van de 1: T I N H TICH PHAN CO BAN B A N G D I N H NGHIA

Phuong phap: Bien doi ham so trong da'u tich phan ve dang tong, hieu cac

ham so c6 the tim dugc nguyen ham va dung djnh nghla de suy ra gia tri

cua tich phan

cos X cos X cosx

Do do: I = 2 J tan' xd(tan x) = -tan' x

Cam nang luyen thi DH - Nguyen ham - Tich phdn - So phirc - Trdn Bd Ha

Bai 11: Tinh cac tich phan sau:

a)C= (sin^xcos3x + cos^xsin3x)ix b) D = cos"xcos4xdx

0 0

Giai a) f(x) = sin^xcos3x + cos^xsin3x

= — (3cosx + cos3x)sin3x + — (3sinx - sin3x)cos3x 4 4

3 3

= — (sin3xcosx + cosSxsinx) = — sin4x Suy ra:

4 4

3 3 C= |f(x)dx = — sin4xdx = cos4x = - A , A o

16 16

) Ta CO cos^xcos4x = f l+cos2x cos4x = -[cos4x + cos4x.cos2xl

= cos4x + —(cos6x + cos2x)

Cam nang luyjn thi DH - Nguyen ham - / / ( / / /)/;.;/; So pln'n Trdn Bd Ha

Ta c6: tanx + cot2x = + = c o s 2 x c o s x + sin2xsin x

cosx sin2x cosxsin2x cos X 1

c o s x s i n 2x s i n 2 x

A ' CI \x _

U o a o r ( x ) = S i n 2 x s i n 3 x s i n 4 x

tan X + c o t 2 x

= — [cos2x - c o s 6 x ] s i n 3 x = — [sinSx cos2x - sin3xcos6x]

= — [sinSx + sinx - ( s i n 9 x + sin(-3x)] = - [sinx + sin3x - sin9x + sinSx]

= J ( l - s i n ^ x)d(sinx)+ J(l-cos^ x)d(cosx)

T a c o : r ( x ) = ; —; = sinx + cosx + cosx - smx = 2cosx

Sin X + cos X sm X + cos x

Cam mifii; In\\n I hi ni! y^uyen ham - Tich/'hchi phirc - J ran JiaTJa'

Bai 19: Cho hiai iiam so: f(x) = 4cosx + 3sinx va g(x) = cosx + 2sinx

a) Tim cac so A va B de g(x) = Af(x) + Bf '(x)

Af(x) + Bf '(x) = 4Acosx + 3Asinx + (-4Bsinx) + SBcosx

= (4A + 3B)cosx + (3A - 4B) sinx

[•4A + 3B = 1 2 „ 1 Dong nhat voi g(x) ta co: J 3 ^ _ 4 Q ^ 2 ^ 5 5

+ Xet dau f(x) tren [a; b]

+ Dung tinh chat phan doan cua tich phan roi tinh tich phan tren tiing doan

Cdm nang luy^n thi DH - Nguyen ham - Tich phdn - So phuc - Trdn Bd Ha

Bai 2: Tinh I = f Vf — sin 2xdx

I = sin x + cosX - (sin x + cos x) "'^ = - 2

Bai 3: Tinh I = f Vtan' x + cot^ x - 2dx

3

Vay I = 4

I I / 2 Bai 5: Tinh I = J (Vl + cos2x - V l - c o s 2 x j d x

Giai sinx|)dx

Cam nang luyjn thi DH - Nguyen ham - Tich phan - S6 phuc ~ TrAr, Ba Ha

Giai

Ta C O cac truong hop sau:

Ne'u a < 0 vi 0 < x < 1 => x - a > 0 khi do 1(a) = J(x=-a.x)dx = y - ^

Phan nua duong thang y ~ ^ ~ ^ ^^^^

Bai 12: Tinh I(m) = | x ' - 2x + m|dx

1

Giai

Cam nang luy^n thi DH - Nguyen ham - Tich phdn - S6 phuc - Trdn Bd Ha

Ta CO cac trirong hop sau:

Phuomg phap: Xet da'u hieu so' f(x) - g(x) tren [a, b] de xac djnh max hoac min

roi diing tinh cha't phan doan de tinh

Bai 1: Tinh I = max x^ + l ; 4 x - 2 dx

Giai

Xet g(x) = x2 - 4x + 3 tren [2; 4] ta c6:

1 2 f(x) + 0 —

Do do: max{x2; 4x - 3| = x^ tren [3, 4]

max|x2; 4x - 3| = 4x - 3 tren [2, 3]

• r f i ,

73

Cdm nang luy^n (hi DM - Nguyen ham - Tich phdn - So phirc ~ Tran Bd Ha

B a i l : Tinh cac tich phan sau:

Cdm nang luy^rt thi DH - Nguyen ham - Tich phdn - So phirc - Tran Bd Hit

-(e^ - i f +-(e^ -1) ,3/2 -(e^ -\f^e^ - 1 5

5 9

- i ; i / 2

J - i t / 2 45 sin^ X sin^ x

e e e Bai 5: Tim K de: J[(K - 4x)dx = 6 - 5K

T \ ; / / / II, my, I men iJii HI I - \yii\\'n Ih'iiii I'ich phdn - So phirc - Iran Bd lid

Bai 10: Tinh cac tich phan sau:

a) I = [(e""' + cos x)cos xdx b)J =