C H U O N G III: N G U Y E N H A M , T I C H P H A N V A U N G D U N G

§ 1 . N G U Y E N H A M

A. K I E N T H U C CO B A N - Nguyen ham: Cho K la mot khoang (a;b), nua khoang (a;b], [a,b) hay doan [a;b]. Ham so F(x) goi la mot nguyen ham ciia ham so f(x) tren K niu: F'(x) = f(x), Vx e K Neu F(x) la mot nguyen ham cua f(x) thi hp cac nguyen ham ciia f(x) la: Jf(x)dx = F(x) + C, C la hang so bat k i

- Bang cac nguyen ham: dx = x + C

Voi a * - 1 ta co:

-dx = In | x | + C

Jkdx = kx + C voi k la hang so

f— dx = In | u | +C J u

..a+1 .a+1 ca.dx + C

a + 1 cosxdx = sinx + C

sinxdx = -cosx + C

dx

dx = •+C a + 1

cosu.u'.dx = sinu + C

cos2 x dx

= tanx + C

cotx + C sm" x exdx = e* + c

i xdx In a

+ c

fu" .u '

f J"sinu.u'.dx =-cosu + C

f u ' —5—dx = tanu + C

Jcos u f—^5— dx = -cotu + C

J s i n u [e u .u 'dx = eu +c

r a u

a u .u'.dx = + c ( a > 0 , a * l ) J In a

- Tinh chat co ban: Neu f va g la hai ham so lien tuc tren K thi: J(f (x) + g(x))dx = Jf(x)dx + [g(x)dx

Jkf(x)dx=kjf( x)dx voi moi so k

- Phuong phap doi bien so: Dang 1: Neu x = u(t) co dao ham lien tuc tren K thi:

J f (x)dx= Jf(u(t)).u'(t).dt

Dang 2: Neu t = v(x) co dao ham lien tuc tren K va co f(x)dx = g(t)dt thi: Jf(x)dx - Jg(t)dt

-BDHSG DSGT12/2-

- Phucmg phap nguyen ham tung phan: Neu u(x), v(x) co dao ham lien tuc tren K thi Judv = uv - Jvdu

B. PHAN DANG TOAN DANG 1: BINH NGHlA VATTNH CHAT

- Be chung minh F(x) la mot nguyen ham ciia f(x) ta chiing minh F'(x) = f(x) voi moi x thuoc D.

- Sir dung tinh chat chit co ban: Neu f va g la hai ham so lien tuc tren K thi:

J(f (x) + g(x))dx = Jf (x)dx + Jg(x)dx

J(f (x) - g(x))dx = Jf (x)dx - J g(x)dx

j"kf (x)dx =k Jf (x)dx voi moi so k

Chii y: - Phoi hop diing bang cong thiic voi cac bien doi chia tach, them bat, khai trien tich so, hang dang thiic, nhan chia lucmg lien hap, viet mu

m phan so v a m = a n

- Voi ham huu t i , neu bac ciia tu Ion hon hoac bang bac ciia mau thi phai chia tach phan da thuc, con lai ham huu t i vdi bac tir be ban mau. Neu bac cua tu be hem bac ciia mau thi bien doi ve tong, hieu ciia cac don thuc va phan thiic don gian bang each dong nhat he so hoac them bat, chia tach,...

it-i—L~) x " - a L ( x - a ) ( x + a) 2a x - a x + a , 1 1 A B hay — r = - - =

x - a" ( x - a ) ( x + a) x - a x + a Vi du 1: Chung minh F(x) la mot nguyen ham ciia f(x):

a)F(x)= l-cotf| + ^l ; f(x) = \2 4) 1 + sinx

b) F(x) = xlnx - x ; f(x) = lnx c) F(x) = e s i n 3 x ; f(x) = 3e s , n 3 x . cos3x

Giai

a)F'(x)= } , = , =-±- =f(x)=>dp „ . 2 x n , TC i 1+smx 2sm + 1-cos x+ 2 4) { 2.

b) F '(x) = lnx + x. — - 1 = lnx = f(x) => dpcm. x

c) F '(x) = e s m 3 x 3cos3x = f(x) => dpcm.

-BDHSG DSGT12/2- 47

V i du 2: Chung minh F(x) la nguyen ham cua f(x):

a) F(x) = ln(x + vl + x2 ) + C

b) F(x) = In

c) F(x) = In

tan— 2 r

C

tan X TC 1

v2 4

1 + a) F '(x)

b) F '(x)

c )F ' (x ) =

V x 2 + 1 _ 1 x + V x + 1 V x + 1

1

; m

; f(x)

+ c ; f(x) =

Giai

dpcm.

v T + ~ ^ i

s inx 1

cosx

2 cos2

1

tan-2 1

x . x smx 2 cos —sin —

2 2 1

2 cos' f x f x - + - tan —

4 u 4 j

=> dpcm.

dpcm.

2 cos rx TCN \ . f x TO

2 cos + — sin —+ — v2 4, 1 U 4)

sin x + rc cosx

V i du 3: Chung minh rang cac ham so sau deu la mot nguyen ham ciia ciing mot ham so, chi ra ham so do:

a) F(x) =

b) F(x)

x 2 +6x + l 2 x - 3 1

va G(x) = x 2 +10 2 x - 3

sm x va G(x) = 10 +cot x.

Giai

a) V i F(x) = X ^ + 6 X ^ + 1 = X 2 + ^ + 3 = G(x) + 3, nen F(x) va G(x) deu la 2 x - 3

mot nguyen ham ciia f(x)

2 x - 3 2x2 - 6 x - 2 0

b) ViG ' (x )

( 2 x - 3 ) 2

2sinxcosx 2cosx

sin x - 1

sin3 x 2 cosx

G'(x) = 2cotx sin" x sm" x

2 cosx

nen F(x), G(x) deu la mot nguyen ham

ciia f(x) s in 3 x

48 -BDHSG DSGT12/2-

V i du 4: Tim nguyen ham ciia cac ham so sau:

a) f(x) = 3x2 + -2

b) f(x) = 2x3 - 5x + 7

c ) f (x) = ( x - l ) ( x 4 + 3x)

a) Jf(x)dx = jf3x2 + -]dx = x3+— +C

d)f (x) = ( x - 9 ) 4

Giai

x 4 5x2

b) | f ( x )dx = J"(2x3 - 5x + 7)dx

c) J(x - l)(x4 + 3x)dx = |(x5 - x4 + 3x2 - 3x) dx

+ 7x + C

x x o 3x _ + x J + C

6 5 2 ( x - 9 ) 5

d) | ( x - 9 ) 4 d x

Vi du 5: Tinh:

a) A = fx(x - 3)1

+ C.

dx b ) B = fx(ax + b f . d x , a ^ 0 .

Giai

a) A = J(x - 3 + 3)(x - 3) 1 1 dx

= }(x-3)12dx + 3|(x-3)ndx= (X"3)13 • (X_3)1Z

b) Ta co B = — f (ax + b)a+1 - (ax + b)a

13 + C

Khi a = - 2 thi B = —

dx

+ In lax + b| + C a2 v. ax + b

Khi a = -1 thiB = —(ax + b-ln|ax + b|) + C

Khi a 5 t - 2 , - 1 thi B = — a

(ax + b ) a + 2

+ (ax + b)° a + 2 a + 1

+ C

V i du 6: Tim nguyen ham ciia cac ham so sau:

a)f (x) = 3 x 2 - \ + -X" X

b) f(x) ( 2 - x ) 2

c) f(x) 2x 4 + 3 d) f (x) = 2x(l - x - 3 ) .

-BDHSG DSGT12/2- 49

) j ( 3 x 2 - ^ + - ] d x = x 3

+ - + ln|x| + C.

Giai 5 3 5

-=- + — dx = x + — x x) x

b) f—L^dx=-^—+ c J ( 2 - x ) 2 2 - x

d) J2x(l - x"3 )dx = J(2x - 2x~2) dx = x2 + - + C .

Vi du 7; Tim nguyen ham cua cac ham so:

a) f(x) = 8x - A b) f(x) = 2 Vx

X4

x3 - 3x2 + 3x - 5 1N c, , x2 - 2x - 3 c) f(x) = — d) f(x) ( x - l ) 2 x - 2

Giai

r r — 8 3

a) Jf(x)dx = j ( 8 x - 2 . x 4 )dx = 4x2 - - x 4 +C 3

b) flz^dx=ff3 + __i J 9 _ V Y J O

7 - 2 1 x , f f„ 1 1 , 1 , dx = 3 x — l n 2 - 7 x +C 2 - 7 x Jl, 2 - 7 x J 7'

c)Tacof(x) = x-l nen ff(x)dx= ~-x + ^- + C ( x - l ) 2 J 2 x - l

3 r x2

d) Ta co f(x) = x nen f(x)dx = 31n|x-2| + C x - 2 ' 2 1

Vi du 8: Tinh: x +1 , , r 2dx

~2 1 } f x ^ L _ d x b) U ™ J ( x - 2 ) ( x + 3) ; J l _ x :

Giai

dx

a) Dat - — ^ =— + J L n e n x + 1 = A(x + 3) + B(x - 2) (x-2)(x+3) x - 2 x + 3

Do do x + 1 = (A + B)x + (3A - 2B), dong nhat he s6 thi:

A + B = 1,3A-2B = 1 nenA= - ,B = -5 5

50 -BDHSG DSGT12/2-

Do do J<v-

x + l

b) j - ^ d X : J l - x

( x - 2 ) ( x + 3)

1 1 •+-

-ax 3 2

• + • 5 ( x - 2 ) 5(x + 3)J dx

: - l n | x - 2 | + - l n | x + 3| + C 5 1 1 5 1 1

dx = In I 1 + x I — In I 1 — x I + C=ln

c) Dat 1 1+x 1-x

A B C ^ L x , . — + — + dong nhat thi co:

1 + x 1-x

x (x + 1) X X- x + 1

f f ( x ) c l x = | f - - + i r + — ldx = - l n | x l - - + hi|x + l | + C = In J \ X X x + l j 1 1 X

x + 1 1 X

d)

Cach khac: Bien doi 1 = x - x - x + x + 1. x4-2 x2-2

= x " ' 3 X - X X - x

= x + -x 2 - 2

x ( x - l ) ( x + l )

Dat — — ? — = - + — + — nen x 2 - 2 = (A + B + C)x2 + (B - C)x x(x-l)(x+l) x x - l x+1

dong nhat he so thi duoc A = 2, B = —. C = —. do do: 2 2

f f ( x ) d x = J l X + • 2 1 1 1 1 x 2 x - l 2 x + 1

1 - x 2 + 21n|x| - — In I x z - 1 I + C.

Vi du 9: Tinh

a) [(Vx+vx)dx b ) f xVx + Vx dx

dx d) j7x( \ /x - Vx + 1) dx

a) J(vx + vx)dx = J f i n

•xVx + Vx V J

Giai

2 3' 2

2 3 1 dx = —x3 + —x3 +C

b) | i W ^ + V x j 3 N

• + X dx = 2Vx - - = + C

d) | v x ( V x - V x " + D dx= J

-BDHSG DSGT12/2-

i A dx = 2 V x - - V x Y + C

2 ( 5 3 1^ X6 -X4 +2x2 6 — 4 - -

dx = — x 6 — x 4 + —x 2 +C 11 7 3

V i du 10: Tinh ck

0 I = f r J V x + 3 - V x - 4

:)E= jVx4 + x"4 + 2dx

b ) J = f ck

d) F = {

Vax+b+Vax+c xdx

, a * 0, b * c

Vx + 2 Giai

I a) 1= ^ | V x + 3+vx^4)dx= 1 j^(x+3) 2 + (x-4) 2

b) J = —|\/ax+b-\/ax+cjdx

dx= 21

3 3 (x + 3)2 +(x-4)2 +C

a (b -c )

\ _ 1 3 1 dx x — c) E = j V ( x 2 + x 2 ) 2 d x = Jj^x2 + \

rx + 2-2dx = f (x + 2)^_2(x + 2) J 37x + 2

= |(x + 2)s -3(x + 2)f +C

Vi du 11: Tim nguyen ham ciia cac ham so:

(ax + b) - ^ ( a x + c) J + C

C. 3

dx

a) f(x) = cos — 2

c) f(x) = tan2x

b) f(x) = sin xdx

d) f(x) = cot2x Giai

f x x cos — dx = 2 sin — + C

2 2 b) jsin2xdx= J 1 -cos2x x sin2x

dx = - - ^ ^ + C 2 2 4

c) f t a n 2 xdx = [ ( — 1 | dx = tan x - x + C J JVcos"x )

d) [cot2 xdx = [( —- 1 ) dx = - cot x - x + C 1 J> sin" x

V i du 12: Tinh:

a) I - |sin3xcos2xdx

c) H = jcos3xdx

b ) J = |cos5xcos7xdx

d)K= j"sin4xdx

Giai

a) I = |(sinx + sin5x)dx = -i(cosx + — cos5x) + C

52 -BDHSG DSGT12/2

b) J - i J(cosl2x + cos2x) dx = ^-(-^sinl2x + sin2x) + C

c) H = — f(cos3x + 3cosx)dx = — sin3x + — sinx + C 4 J 12 4

d) sin4x = — (1 - cos2x)2 = — (1 - 2cos2x + cos22x) 4 4

13 1 = — ( 2cos2x + — cos4x)

4 2 2 13 1

nen K = — (—x - sin2x + — sin4x) + C 4 2 8

Vi du 13: Tinh: \ f 1 i , , rl-cos2x

a) K - s =— dx b) dx J sin xcos x ' cos x

sin4xsin3x , 1N f2x sin2 x - cos2 x + 1 - x , r dx

. r sm^xsincsx , r c) IT 77T dx d)

J t a n x + cct2x J xs in x

Giai

0 f . 9 — ~ ° 1 x = f f — ^ + — | d x = t anx-co tx+ C J Gin" vf i ' sin" x cos" x J V cos" x sm2 x

. rl-cos2x r2sin"x / 1 o) dx= — dx = 2 — 1 dx = 2(tanx-x) + C

J cos x J cos x -"Icos X ' X j sinx cos2x cosx 1

c) tan x + cot zx = h — = : = — nen cosx sin2x cosx sinx sin2x

sin4xsin3x . . 1 . = sin 4x sm 3x sm zx = — sm 4x(cos ox - cos x)

tan x + cot x 2 = (sin4xcos5x - sin4xcosx) = -— (sin9x - sinx - sin5x - sin3x).

Vay f5111^8111^ ^ = _ jL (_ i_ COs9x + cosx + — cos5x + — cos3x) + C J tanx + cotx 4 9 5 3

,v r2xsin2x-cos2 x + l-x , / 1 1 V _ , i i _ d) dx = 2 + — dx = 2x + In | x | + cotx + C

J xsin x J |v x sin xj V i du 14: Tinh: 2 e'

) L * - \ ) d x b) | l 0 2 x d x

c) J ( 2 x - 3 x ) 2 d x d) 5 " X ) d x

53

Giai 2

a) j | e * - — j d x = [(e x +2e--x)dx = ex -2e'x +C

b) fl02xdx=fl00xdx = ^ + C=-^ + C ' J In 100 21nl0

c) f(2x-3x)2dx = (•(4x-2.6x+9x)dx = — -2 — + — + C 1 J V ; In 4 In 6 In 9

f5> x > - - 3 " x

v3y v3y J

dx =

5 T 3X

+ C. In5 ln3

3 V i du 15: Tinh:

\ T _ f COS X f „ a ) 1 _ : dx b ) E = cos2 x.cos3xdx

J cos x +s inx J Giai

a)XetJ= [ dx thi: I + J = fdx = x + C, J cosx + sinx J 1

T T _ fcosx - sinx . j . i - J - : dx = In cosx + smx + C9

J cos x +s inx 1 1

Suy ra I = —(x + In |cos x + sin x|) + C. 2

b)XetF= |sin2x.cos3xdx thi:E + F = [cos3xdx =^-sin3x + C,

E - F = jcos 2x. cos 3xdx = - |(cos 5x + cos x)dx

= — sin 5x + — sin x + C9

10 2 2

Suy raE = -^sin5x+ -sin3x+ -sinx + C. 20 6 4

Vi du 16: Tinh: a) J(l + tanx)2 e2xdx b) [xVdx

Giai

a) J(l + tan)2e2xdx = f(l + tan2x + 2tanx)e2xdx

= J(tanx.e2x)dx =tanx.e2x + C

mm npHj ^g^r

54 -BDHSG DSGT12/2-

b) Xet f(x) = (ax4 + bx3 + cx2 + dx + m)ex thi f'(x) = (4ax3 + 3bx2 + 2cx + d)ex + (ax4 + bx3 + cx2 + dx + m)ex

= [ax4 + (4a + b)x3 + (3b + c)x2 + (2c + d)x + d + m]ex

Ta co f ' ( x ) = x 4 e x <=> a = 1, 4a + b = 0, 3b + c = 0, 2c + d = 0, d + m = 0 oa = l , b = - 4 , c = 12, d = -24, m = 24.

Vay J x V d x = (x 4 - 4x3 + 12x2 - 24x + 24)ex + C.

DANG 2: PHUONG PHAP BIEN B 6 l BIEN S 6

Dang 1: Neu x = u(t) co dao ham lien tuc tren K thi:

j f (x )dx= jf(u(t)) .u ' ( t ) .dt

Dang 2: Neu t = v(x) co dao ham lien tuc tren K va co f(x)dx = g(t)dt thi:

Jf(x)dx = Jg(t)dt

Chii y: - Su dung cac cong thuc mo rong kx voi k * 0. M d rong cong thuc x thanh u kem theo du = u'.dx, luu y dau cong trir va he so nhan chia. - Neu J f(x)dx = F(x) + C thi J f(u)du = F(u) + C - Chon dat bien thich hop, bien u ma bieu thuc co san u '

- Doi bien dai so: u = g(x), x = h(t), t = —, ... x

Dang , 1 thi dat t = x + %/x2 + k ' V x 2 + k

- Doi bien luong giac: Dang 9

1

9 thi dat x = a.tant x" -t- a"

Dang \la2 - x 2 thi dat x = asint hay x = acost

Dang Vx 2 + a2 thi dat x = atant.

Dang Vx 2 - a 2 thi dat x = a

sin t , x

Bien doi theo goc phu t = tan— 2

Bac le voi sinx thi dat t = cosx, bac le vdi cosx thi dat t = sinx

Dac biet d6i bien t = - x , t = TC - x, t = - x.

- Nguyen ham lien ket, de tinh I thi dat them J ma viec tinh I +J va I - J, I + kJ va I - mJ thuan loi hon, tu do suy ra I

-BDHSG DSGT12/2- 55

V i d u 1: Tinh:

a) f(2x + l ) 9 dx

c) j"x3(l + x4)3dx

b) jx(3 - x)5dx

d) f x

Giai

18 dx

a) Doi bien u = 2x + 1 thi du = 2dx => dx = — du 2

f(2x + 2 ) 9 dx= - f u 9 d u - . — . u 1 0 + C = — ( 2 x + l ) 1 0 + C J 2 J 2 10 20

Cach khac: Viet gop

j(2x + 2)9dx = - | (2x + l ) 9 d(2x +1) = JL(2x + l ) 1 0 + C

b) Dat u = 3- x=>x = 3- u=>dx = -du

Jx(3 - 5)5dx = - J(3 - u) .u 5 .du = | ( u 6 - 3u 5 )du

= i u 7 - 3 - ' 7 6

u° + C = ( 3 - x ) 6 1 + C

c) f ( x 3 ( l + x 4 ) 3 d x = i J ( l + x 4 ) 3 .d(l + x 4 ) = ^ ( 1 + x 4 ) 4 + C

x 3 x 2

d) Dat u = 1 ^> du = — dx => x 2 dx = 6du 18 6

f 18

V i du 2: Tinh

2x + l

57 dx = 6 f u 5 7 d u = — u 5 8 +C

29 29 ,18

58

a) I - - dx J X + X + 1

d) J - d x

b ) J .

2 2 x - a

J x 3 +

d + x 2 ) 2

d x _ x 5

Giai

dx

+ C

:) f -J x ( l + x a )

J X + X + 1 dx

a) Doi bien: u = x 2 + x + 1 thi du = (2x + l)dx

2x +1 . r d u f - ^ - d x = f — dx = l n u + C = ln(x 2 + x + l ) + C

J X" + x + 1 1 u

56 -BDHSG DSGT12/2-

b) Dat u = 1 + x 2 thi du = 2xdx => xdx = - d u

1 rdu 1 1 - . - + C

1 C '(1 + x 2 ) 2 " " 2 - V 2"u 2(1+ x 2 )

, f dx _ r x7dx 1 r d(x8) _1. x8

°j •'x(l + x8)"Jx8(l + x8)"8 Jx8(l + x8)"8 "l + x8 +

dx 1 d ) f - r ^ r ^ f l — - —

J x - a" 2 a J > , x - a x + a

ff

1 , 1 , x - a dx = — I n

) 2a x + a + c

1 - x 2

1 + x^ dx

-3 1 X x A - - +

X 1 + X 1

- 1 V l + x" dx = — - + In —r-:— + C

2x2 x

d x + In x - x + 1

x z + X + 1 + C

V i du 3: Tinh:

a) [s in 4 xcos xdx

c) J— COi - - 1

b) j"xcos(x2)dx

d) f^sin —cos —dx J x" x x

Giai a) Dat t = sinx thi dt = cosx dx

[sin4 x cos xdx = [t4dt = — t5 + C = Sm X + C . J J 5 5

b) Dat t = x2 thi dt = 2xdx

fx cos(x2 )dx = — ("cos t.dt = — sin t + C = — sin(x2) + C J 9 J 9 9.

c) Dat t = - - 1 thi dt = —r- dx => -4- dx = -d t

j-^cos^" 2 " - 1 jdx = - Jcost.dt = - s int + C = - s i n ^ — - 1 j + C

d )Dat t = s i n - ^ > d t x

1 1 A —rr.COS— dX

-BDHSG DSGT12/2- 57

Do do: f—5-sin— cos—dx = - [t.dt = - ^ t 2 + C = - - s i n 2 — + C J x 2 x x J 2 2 x

V i du 4: Tinh: f 3 >\ X2 +1 a) A = | x 2 sin

c) M = jsin3 xcos4 xdx

dx b ) B k i

cosx 2 sin x

-dx

X T rs in 3 x . d ) N = :—dx

COS X

3 i Giai

I a) Dat u = x 2 + 1 => du = —x2dx => x 2 dx = — du

2 3 z r 2 2 -

A = — sinudu = —cosu + C = — c o s f x 2 + 1) + C 3 J 3 3

1 rdu I , , 1 , . , - In u + C = — In 1 + 2 sm x + C .

I 2 I I

2 r 2 — cosu + L = — 3 3

b) Doi bien: u = 1 + 2sinx thi du = 2cosxdx

B = -f-2 •> u 2

c) Dat u = cosx thi du = -sinxdx M= J(l -cos2 x)cos4 x. sin xdx

= -f(l-u2)u4.du= f(u6-u4)du=1u7--u5+C= -cos7x--J J 7 5 7 5

d) Dat u = cosx thi du = -sinxdx

cos5 x + C.

K - f (1 —cos" x)s inx

cos4 x - J l u 4 u 2

d u = - u - 3 - i 1 3

C

du

1 3cos x cosx

• + C

V i du 5: Tinh: 1

J cm13 Y

>J

sin x

sin x 3/ ~ vi—

dx cos X

1 + sinx dx + cos X

sinxcosx

va sm" x + b cos x Giai

dx (a2 5t b 2 )

a) Dat u = cosx thi du = -sinxdx

r 1 dx= rsmxdx = _r J sin x J v

du s i n 4 x J ( l - u 2 ) 2

58 -BDHSG DSGT12/2

Ta co: 1 1 _ 1 [(u + l ) - ( u - l ) f

(1-u2)2 (u-l)2(u + l)2 4' (u-l)2(u + l)2

f 1 1 • + -

nen : [— \— dx = — In J s in 3 x 9

b) Ta co: 1 + sin x

(u + 1)2 ( u - 1 ) 2 u 2 - l ,

cosx tan — 2 2 sin x

• + C

x

1 + COSX o 2 X X ZCOS COS

2 2 c) Dat u = cosx thi du = -sinxdx

sm 1 o A f l + sinx , x

+ nen dx = tan — J 1 + cos x 2

{ ^ ^ d x = - J - ^ L = - f u ^ d u = -3u3 + C = - S 3 / ^ + ' 3/ 2 VCOS X

d) Dat u = Va2 sin2 x + b2 cos2 x => du = (a2 - b 2 ) sin xcosx

4 2 - 2 i 2 2 a sin x + b cos x

dx

f smx cosx -dx

Va2 s i n 2 x + b 2 cos2 x a2 + b 2 J a 2 - b = 2 \ 2 f d u = 2 1 , 9 u + c

1 / 2 • 2" r~2 9 ^ —5 „-va sin x + b cos x + C a - b 2

V i du 6: Tinh: a) jcot xdx b) j"tan3xdx

:) fcot 5 xdx d) f — J J 1 +

cotx s in 9 x

dx

Giai a) Dat u = sinx thi du = cosxdx

fcotxdx = f^^dx = [— du = ln|u| + C = ln|sinx| + C J J sm x J u 1 1

b) Jtan3 xdx = [tan(l + tan2 x - l)dx

fx 1 x rsinx | 1 Q 1 . - t a n x d x ( t a n x ) - dx = — tan x + In cosx + C

J J cosx 2 1 1

. r E , rcos5 x , r(l -sin2 x)2 . c) cot5 xdx = r—dx= — U ( s i n x )

> J sin x } sin 0 x 2 1

—r, 1 sm x sm x smx d(sinx) = - 1

4sin x sin~x + ln|sin3

-BDHSG DSGT12/2-

d) M = i - d * = J -J 1 + sm x J s

cosx.sin x dx = ^ I k d(sin9 x)

' sin9 x ( l + sin9 x) 9 J sin9 x ( l + sin9 x)

1 rd(sin9x) 1 rd(sin9x) _ 1 F 9 J sm9 x 9

Vidu7: Tinh:

fd(sm x) •> sin9 x + 1

- I n 9

sin9 x 1 + sin x

+ C

a ) A = JxV7-3x 2 dx

c)Q= f-^rdx

J1 - Vx

b ) B = j " (2x - 3)Vx~r3dx

d)P= j Vx 2 +4

Giai a) D6i bien: t = 7 - 3x2 thi dt = -6xdx

A = - - f y t . d t = - - t 2 + C = - - ( 7 - 3 x 2 ) 2 + C 4 ' 9 3

b) Dat t = V x - 3 => x = t 2 + 3 => dx = 2t.dt

B = 2 |(2t2 + 3)t2dt = 2 J(2t4 + 3t2)dt

= -t5+2t3+C = -(x-3)l(2x-l) + C. 5 5

c) Dat t = 1 - Vx" => x = (1 - t)2 => dx = -2(1 - t)dt

Q=2jl-_idt = 2^1-ijdt

= 2(t-ln|t|) + C = -2(Vx + ln| 1 - Vx |) +C i 1 3 2

d ) P = f _ ^ = d x = - f ( x 2 +4 ) 3d(x2+4) = - ( x 2 + 4 ) 3 + C J 3/_ 1 . A 9. J 4 , 3 / x 2 + 4

Vi du 8: Tinh: dx

o f

;)f

/ x ( l + V x ) 2

dx , dx

b) f £ ± £ « k J X

Vx 2 +9 Giai

(x + l )dx x(l + xex)

a) Dat u = 1 + Vx

dx

du 2Vx"

dx

f V x ( l + V x ) 2 J u" u + C =

1 +Vx c

60 -BDHSG DSGT12/2-

b) Dat t = V l + x

rvT

x = t 2 - 1 => dx = 2t.dt

1 + dt

2 [dt + ~ ) d t = 2 t + l n I* _ 1 l " + !| + C

2VITl: + ln^^^+C. V l + x +1

dx dt

V l + x +1

c) Dat t = x + Vx2 +9 =>dt =

dx rdt r ax _ rdt KL2 J-Q ~ J t ' V x 2 + 9

d) Dat t = 1 + xex thi dt = (x + l)exdx

>(x + l)dx rf 1 1

1+ , dx^- , V x 2 + 9 j V x 2 + 9 t

ln|t| + C = ln x +Vx2 +9 +C

r(x + l)dx _ r 1 1 ' x ( l + xe x ) ~~ Jl t - 1 t t ( l + xe x )

Vi du 9: Tinh:

a) jxe~x dx

r(lnx)~

t - 1 dt = ln

t - 1 t

C = In xe' 1 + xex

+ C

dx

b) j*es i n xcosxdx

d) f dx 'e - e

Giai a) Doi bien: u = - x 2 thi du = -2xdx

1 1 2 V + c = - - e - x + c j"xe x"dx = -— jV 'du

b) jVmxcosxdx = |esmxd(sinx) = esmx +C

C ) j C l ^ O l d x = | ( l n x ) 2 d ( l n x ) = 1 l n 3 x + c

d) Dat t = ex thi dt = exdx => dx = -dt t

J e

x - e _ x J t ( t - t _ 1 ) J t 2 - 1 2 ^ t - l t + 1. Idt

= i ( l n | t - l | - l n | t + l | ) + C = - l n 2 2

e" - 1 ex +1

+ C

-BDHSG DSGT12/2-

V i du 10: Tim ham so y = f(x) biet: a)dy = 12x(3x 2 - l ) 3 dx v a f ( l ) =

7 3 b) y' = sin x. cos x va f(n) = 79.

a) Dat u = 3x2 - 1 thi du = 6xdx nen 12xdx = 2du. Giai n 12x ( 3 x 2 - l ) 4

f ( x ) = Jdy = |2u 3 du = - y + C = v d X ~ i J +C

Vi f(t) = 8 + C = 3 nen C = -5. Vay f(x) = (3x~~1)4 - 5 . 2

b) Dat t = sinx thi dt = cosxdx f ( x ) = j"y 'dx= j s i n 7 x ( l - s i n 2 x) cos xdx = j V ( l - t 2 ) d t

s i n 8 x s i n 1 0 x = f ( t 7 - t 9 ) d t = - L - ^ + C = ̂ - ^ ^ + C

J 8 10 8 10 8 • 10

R1TY Y sin V V i f(Tt) = 79 nen C = 79. Vay f(x) = - + 79 .

Vi du 11: Dat In = jtan"xdx , n e Tinh L, theo In_2, n > 3.

Giai V d i n > 3 : I n = [ t a n n _ 2 x tan2xdx = J"tann"2x(tan2 x + 1 - l)dx

= ("tan11-2 xdx (tanx) - In_2 = —-— tan11"1 x - In_2. J n - l

DANG 3: NGUYEN HAM TUNG PHAN

Neu u(x), v(x) co dao ham lien tuc tren K thi Judv = uv - Jvdu

Chu y: - Chon dat u va dv de dua ve nguyen ham gan cong thuc hon hoac dang vong tron, tung phan lien tiep,... - Cac dang ham hon hop e\sinx, excosx, P(x).ex, x.lnx, x2.lnx, P(x).lnx,... deu dung phuong phap tirng phan, co the dat u, dv lien tiep nhidu lan. - Voi tich phan truy hoi I n theo I n - i hay In_2 thi sin1, x, cos" x tach luy thira 1 va dung phuong phap nguyen ham tirng phan con tan" x, cot" x tach luy thua 2 va dirng phuong phap nguyen ham doi bien so.

Vi du 1: Tinh: a) fxcosxdx b) [—\— dx J J sin x

Giai a) Dat u(x) = x, v'(x) = cosx. Khi do u'(x) = 1, v(x) = sinx

Ta co jxcosxdx = judv = uv - jvdu

= xsinx - Jsinxdx = xsinx + cosx + C

62 -BDHSG DSGT12/2-

b) I——*— dx = - (xd(cot x) = -xco tx + (cot xdx Jsm x J J

rcos x i i = -xcotx + — dx = - x c o t x + In sinx + C

J sinx Vi du2 : Tinh:

a) I = Jx2cos2xdx b) J = JsinVxdx

Giai

a) Dat u = x2, v' = cos2x. Khi do u' = 2x, v = — sin2x. 2

I = ^x2sin2x- J"xsin2xdx = ^x2sin2x + i Jxd(cos2x)

= — x2sin2x + — xcos2x - — sin2x + C. 2 2 4

b) Dat t = Vx => x = t2 => dx = 2tdt J= Jsmt.2t.dt = 2jt.sint.dt = 2(-tcost + Jcostdt)

= 2(-tcost + sint) + C = 2(—Vx cos Vx + sin Vx ) + C Vi du 3: Tinh: a) Jlnxdx b) JVx.lnxdx

Giai

a) Dat u = lnx, dv = dx. Khi do du = — dx, v = x. Ta co: x

Jlnxdx = xlnx - Jx. —dx = xlnx - Jdx = xlnx - x + C.

i— 1 2 -b) Dat u = lnx, v ' = v x ^ > u ' = — . v = — x 2 Taco:

x 3 fVxlnxdx = — x2 lnx - f— x2dx = —x2 lnx x2 +C

J 3 J 3 3 9

Vi du 4: Tinh: a) fx3 ln(2x)dx b) fxln—^—dx J J 1 + x

Giai

a) Dat u = ln(2x), v' = x3 Khi do u' = -. v = — x 4

Jx3 ln(2x)dx = Judv = uv - Jvdu

11 4 = - x 4 l n ( 2 x ) - - f x 3 d x = i x 4 l n ( 2 x ) - — + C.

4 4 J 4 16

-BDHSG DSGT12/2-

x 1 x 2

b) Bat u = In . du = xdx. Khi do du = . v = — 1 + x x ( l + x) 2

fx In —X—dx = — In — — f X dx J 1+x 2 1 + x 2 J l + x

- x21 x 1 / 1 V x2, x 1, ,, | 1 n

In + - 1 dx = — I n + - l n 1 + x — x + C 2 1 + x 2 - V + x ) 2 1 + x 2

Vi du 5: Tinh: a) I = fxexdx b) J = jx3.exdx

Giai a) I = fxexdx = xex - jVdx = xex - ex + C = ex(x - 1) + C

b) Dat u = x3, v' = ex thi J = Jx3.exdx = exx3 - 3 Jx2.exdx

Dat u = x2, v' = ex thi fx2.exdx = x2ex - 2 jxexdx = x2ex -21

Do do J = ex(x3 - 3x2 + 6x - 6) + C.

Vi du 6: Tinh: a) A = fe^^dx b) B = jln(x + Vl + x2 )dx

Giai Q

a) Dat t = V 3 x - 9 => 3x = t 2 + 9 => dx = - t d t 3

A = | jte'dt. Dat u = t, v' = el thi Jte'dt = e' - e' + C

2 n e n A = | ( V 3 x - 9 e ^ - e 7 3 ^ j + C

v = x. b) Dat u = ln(x + V l + x 2 ), dv = dx khi du = - r = i = V l + x 2

B = xln(x + Vl + x2)-J dx = xln(x + Vl + x2)-Vl + x2 +C

Vi du 7: Tinh: a) fln(sinx) dx b) fXsin2xdx

/ l + x z

•ln(sinx) cos2 x Giai Tn(sinx) cos2 x

. r irnsinx; r. . , i) T, dx - ln(sinx) d(tanx)

* rric v J = tanx.ln(sinx) - [dx = tanx.ln(sinx) -x + C.

b) Dat u = x, dv = sin xdx = — (1 - cos2x)dx

Khi do du = dx, v = — (x - — sin2x) 2 2

-BDHSG DSGT12/Z-

jxs in 2 xdx = Judv = uv - jvdu

= —(x- — sin2x)x - — jj x - — sin2x jdx 2 J l . 2 J

= —fx2 - — sin2x) - — (— + —cos2x) + C 2 2 2 2 4

- x2 x • o 1 o n sm 2x cos 2x + L

4 4 8 Vi du 8: Tinh: a) I = jsin(lnx)dx b) J = |ex (cosx + 2xsinx)dx Giai a) Dat u = lnx thi x = eu nen dx = eudu

A = jsinu.eudu = |sinud(eu)= sinu.e" - jcosu.eudu

= sinu.e" - Jcosu .d(eu) = sinu.e11 - cosu.eu - Jsinu.eMu

Tu do suy ra A = — x(sin(lnx) - cos(lnx)) + C 2

b) Dat u = e"2, dv = cosx. Khi do du = 2x ex' dx, v = sinx

jex .cosxdx = ex .sinx-J2xex .sinxdx

nen J= j"ex (cosx + 2xsinx) dx = ex .sinx + C

Vi du 9: Dat In = [ sinnxdx, n e N Tinh In theo In_2 va n > 3.

Giai

In = Isin""^.sinxdx = — |sinn_1xd(cosx)

= -sinn_1x.cosx + (n - 1) jsinn"2x.cos2xdx

= -sinn_1x.cosx + (n - 1) |sinn_2x(l - sin 2x)dx

= -sinn_1x.cosx + (n - 1 )In-2 - (n - l)In

Do do In = -—sin11-1 x.cosx + ——-In_2 • n n

Vi du 10: Dat I„ = fxnexdx , n e N* Tinh I theo In_i voi n > 2. Suy ra I3.

Giai

In = Jx"d(ex) = xn.ex - n fx^e'dx = xn.ex - nl„_i

Do do I3 = x3ex - 3I2, h = x2ex - 21,, I, = |xexdx = ex(x - 1) + C

nen I3 = ex(x3 - 3x2 + 6x - 6) + C.

-BDHSG DSGT12/2-