200 Bai Tap Tich Phan Megabook

7/25/2019 200 Bai Tap Tich Phan Megabook

1/48

Trang 1

TP1: TCH PHN HM S HU T

Dng 1: Tch phn thc

Cu 1.x

I dx

x x

2 2

21 7 12

I dxx x

2

1

16 91

4 3

= x x x

2

116ln 4 9ln 3 = 1 25ln2 16ln3 .

Cu 2.dx

I

x x

2

5 31

Ta c:x

xx x x x3 2 3 2

1 1 1

( 1) 1

I x xx

2

2

21 1 3 1 3ln ln( 1) ln2 ln5

2 2 2 812

Cu 3.x

I dx

x x x

5 2

3 24

3 1

2 5 6

I

2 4 13 7 14ln ln ln2

3 3 15 6 5

Cu 4.xdx

I

x

1

0 3( 1)

Ta c:x x

x xx x

2 3

3 3

1 1( 1) ( 1)( 1) ( 1)

I x x dx 1 2 3

0

1( 1) ( 1) 8

Dng 2: i bin s

Cu 5.x

I dx

x

2

4

( 1)

(2 1)

Ta c:

x xf x

x x

21 1 1

( ) . .3 2 1 2 1

x

I Cx

31 1

9 2 1

Cu 6.

xI dx

x

991

1010

7 1

2 1

x dx x xI d

x x xx

99 991 1

20 0

7 1 1 7 1 7 1

2 1 9 2 1 2 12 1

x

x

1001001 1 7 1 11 2 1

09 100 2 1 900

Cu 7.

xI dx

x

1

2 20

5( 4)

t t x2 4 I

1

8

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2/48

Trang 2

Cu 8.x

I dx

x

1 7

2 50 (1 )

t t x dt xdx 21 2

tI dt

t

2 3

5 51

1 ( 1) 1 1.

2 42

Cu 9. I x x dx 1

5 3 6

0

(1 )

tdt t t

t x dt x dx dx I t t dt

x

1 7 83 2 6

20

1 1 11 3 (1 )

3 3 7 8 1683

Cu 10. I dxx x

4 3

41

1

( 1)

t t x

2 t

I dtt t

3

21

1 1 1 3ln

2 4 21

Cu 11.dx

I

x x

2

10 21 .( 1)

x dx

I

x x

2 4

5 10 21

.

.( 1)

. t t x

5 dt

I

t t

32

2 21

1

5 ( 1)

Cu 12.x

I dx

x x

2 7

71

1

(1 )

x xI dx

x x

2 7 6

7 71

(1 ).

.(1 )

. t t x

7 t

I dtt t

128

1

1 1

7 (1 )

Cu 13.dx

I

x x

3

6 21 (1 )

t : xt

1

tI dt t t dt

t t

3

1634 2

2 21 3

3

11

1 1

=

117 41 3

135 12

Cu 14.x

I dx

x

2 2001

2 10021

.(1 )

x

I dx dx

x x

xx

2 22004

3 2 1002 10021 1

32

1. .

(1 ) 1 1

. t t dt dx x x

2 3

1 21 .

Cch 2: Ta c:x xdx

I

x x

1 2000

2 2000 2 20

1 .2

2 (1 ) (1 )

. t t x dt xdx

21 2

t

I dt d t tt t

10002 21000

1000 2 10011 1

1 ( 1) 1 1 1 11 1

2 2 2002.2

Cu 15.x

I dx

x

2 2

41

1

1

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3/48

Trang 3

Ta c:x x

xx

x

2 2

42

2

11

1

11

. t t x dt dx x x

2

1 11

dt

I dt

t tt

3 3

2 2

21 1

1 1 1

2 2 2 22

t

t

31 2 1 2 1

.ln ln22 2 2 2 2 2 11

Cu 16.x

I dx

x

2 2

41

1

1

Ta c:x x

x xx

2 2

4

22

11

1

11

.t t x dt dx x

x

2

1 11

dt

I

t

5

2

2

2 2

.

tdu

t u dt

u2

2 tan 2cos

; u u u u1 25 5

tan 2 arctan2; tan arctan2 2

u

u

I du u u

2

1

2 1

2 2 2 5( ) arctan arctan2

2 2 2 2

Cu 17.x

I dx

x x

2 2

3

1

1

Ta c: xI dx

xx

2 2

1

11

1

. t t xx

1 I

4ln

5

Cu 18.x

I dx

x

1 4

60

1

1

Ta c:x x x x x x x x

x x x x x x x x

4 4 2 2 4 2 2 2

6 6 2 4 2 6 2 6

1 ( 1) 1 1

1 1 ( 1)( 1) 1 1 1

d x

I dx dx

x x

1 1 3

2 3 20 0

1 1 ( ) 1.

3 4 3 4 31 ( ) 1

Cu 19.x

I dx

x

3

23

40 1

x

I dx dx

x x x x

3 3

23 3

2 2 2 20 0

1 1 1 1ln(2 3)

2 4 12( 1)( 1) 1 1

Cu 20.xdx

I

x x

1

4 20 1

. t t x2

dt dt I

t t

t

1 1

2 220 0

1 1

2 2 6 31

1 32 2

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4/48

Trang 4

Cu 21.x

I dx

x x

1 5

22

4 21

1

1

Ta c:x x

x xx

x

2 2

4 22

2

111

11 1

. t t x dt dx x x

2

1 11

dt

I

t

1

20 1

. tdu

t u dt

u2

tancos

I du4

0 4

TP2: TCH PHN HM S V T

Dng 1: i bin s dng 1

Cu 22.x

I dx

x x23 9 1

xI dx x x x dx x dx x x dx

x x

2 2 2

2(3 9 1) 3 9 1

3 9 1

+ I x dx x C 2 31 13 + I x x dx 2

2 9 1 x d x x C 3

2 2 2 22

1 19 1 (9 1) (9 1)

18 27

I x x C

32 32

1(9 1)

27

Cu 23.x x

I dx

x x

2

1

x x

dx

x x

2

1

x xdx dx

x x x x

2

1 1

.

+ x

I dx

x x

2

1

1

. t t= x x t x x 21 1 x t3 2 2( 1) x dx t t dt 2 2

4( 1)

3

t dt t t C 2 34 4 4

( 1)3 9 3

= x x x x C3

1

4 41 1

9 3

+x

I dx

x x2

1

=

d x x

x x

2 (1 )

3 1

= x x C2

41

3

Vy: I x x C 3

41

9

Cu 24.

xI dx

x

4

0

2 1

1 2 1

tt x

2 1

. I =

tdt

t

3 2

12 ln21 .

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5/48

Trang 5

Cu 25.dx

I

x x

6

2 2 1 4 1

t t x4 1 . I

3 1ln

2 12

Cu 26. I x x dx 1

3 2

0

1 t: t x21 I t t dt

12 4

0

2

15 .

Cu 27.x

I dx

x

1

0

1

1

t t x dx t dt2 . . I =t t

dtt

1 3

0

21

= t t dt

t

12

0

22 2

1

=

114ln2

3 .

Cu 28.x

I dx

x x

3

0

3

3 1 3

t t x tdu dx 1 2 t t

I dt t dt dt tt t

2 2 23

21 1 1

2 8 1(2 6) 6

13 2

33 6 ln

2

Cu 29. I x x dx 0

3

1

. 1

t

t tt x t x dx t dt I t dt

11 7 4

3 2 33

00

9

1 1 3 3( 1) 3 7 4 28

Cu 30.x

I dx

x x

5 2

1

1

3 1

ttdt

t x dx 2

3 13

t

tdtI

tt

22

4

22

11

3 2.

31.

3

dtt dt

t

4 42

22 2

2( 1) 2

9 1

tt t

t

3 4 42 1 1 100 9ln ln .9 3 1 27 52 2

Cu 31.x x

I dx

x

3 2

0

2 1

1

t x t x t21 1 dx tdt 2

t t t

I tdt t t dt t t

22 22 2 2 5

4 2 3

11 1

2( 1) ( 1) 1 4 542 2 (2 3 ) 2

5 5

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6/48

Trang 6

Cu 32.x dx

I

x x

1 2

0

2( 1) 1

t t x t x tdt dx 21 1 2

t tI tdt t dt t

t tt

222 22 2 3

311 1

( 1) 1 1 16 11 2.2 2 2 2

3 3

Cu 33.

xI dx

x

4

20

1

1 1 2

tdx

t x dt dx t dt

x

1 1 2 ( 1)1 2

vt t

x

2 2

2

Ta c: I =t t t t t t

dt dt t dt tt t t

4 4 42 3 2

2 2 22 2 2

1 ( 2 2)( 1) 1 3 4 2 1 4 232 2 2

=t

t tt

21 23 4ln

2 2

=1

2ln24

Cu 34.x

I dx

x

8

23

1

1

x

I dx

x x

8

2 23

1

1 1

= x x x8

2 23

1 ln 1

= 1 ln 3 2 ln 8 3

Cu 35. I x x x dx 1

3 2

0

( 1) 2

I x x x dx x x x x x dx 1 1

3 2 2 2

0 0

( 1) 2 ( 2 1) 2 ( 1) . t t x x22 I

2

15 .

Cu 36.x x x

I dx

x x

2 3 2

20

2 3

1

x x xI dx

x x

2 2

20

( )(2 1)

1

. t t x x

2 1 I t dt 3

2

1

42 ( 1)3

.

Cu 37.x dx

I

x

2 3

3 20 4

t t x x t xdx t dt 3 2 2 3 24 4 2 3 I t t dt 3

24 3

4

3 3 8( 4 ) 4 2

2 2 5

Cu 38.dx

I

x x

1

211 1

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7/48

Trang 7

Ta c:x x x x

I dx dx xx x

1 12 2

2 21 1

1 1 1 1

2(1 ) (1 )

xdx dx

x x

1 1 2

1 1

1 1 11

2 2

+ I dx x x x

11

1 11

1 1 11 ln | 12 2

+x

I dxx

1 2

21

1

2

. t t x t x tdt xdx

2 2 21 1 2 2 I2=t dt

t

2 2

22

02( 1)

Vy: I 1 .

Cch 2: t t x x2 1 .

Cu 39. x x

I dx

x

1

3 31

41

3

Ta c: I dx

x x

11

3

2 31

3

1 11 .

. t tx

2

11 I 6 .

Cu 40.x

I dxx

2 2

1

4

Ta c:x

I xdx

x

2 2

21

4 .t t = x t x tdt xdx

2 2 24 4

I =t tdt t t

dt dt t tt t t

00 0 02

2 2 2

33 3 3

( ) 4 2(1 ) ln

24 4 4

=

2 33 ln

2 3

Cu 41.x

I dx

x x

2 5

2 22 ( 1) 5

t t x2 5

dtI

t

5

23

1 15ln

4 74

.

Cu 42.x

I dx

x x

27

3 21

2

t t x6 t t

I dt dt tt t t t

3 33

2 2 21 1

2 2 2 15 5 1

( 1) 1 1

2 55 3 1 ln

3 12

Cu 43. I dxx x

1

20

1

1

t t x x x 2 1 dt

I tt

1 31 3

11

2 3 2 3ln(2 1) ln

2 1 3

Cu 44.x

I dx

x x

3 2

2 20 (1 1 ) (2 1 )

t x t2 1 I t dt t t

4

23

42 36 4

2 16 12 42ln 3

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8/48

Trang 8

Cu 45.x

I dx

x x x x

3 2

0 2( 1) 2 1 1

t t x 1 t t dt

I t dt t t

2 22 22

21 1

2 ( 1)

2 ( 1)( 1)

t2

3

1

2 2

( 1)3 3

Cu 46.x x x

I dx

x

32 2 3

41

2011

Ta c: xI dx dx M N x x

32 2 2 22

3 31 1

11

2011

xM dx

x

32 2

23

1

11

.t tx

32

1 1 M t dt

37

323

0

3 21 72 128

N dx x dx

x x

2 22 2 2 23

3 21 1 1

2011 2011 140772011

162

I314077 21 7

16 128 .

Cu 47.dx

I

x x

1

33 30 (1 ). 1

t t x3 31 t dt

I dt

t t t t

3 32 22

2 21 14 3 2 33 3.( 1) .( 1)

dt dt tdt

t

tt ttt

3 3 3

2

3

2 2 2 3

2 2 41 1 1

3 342 333

11

111. 1

tdt

u du

t t3 4

1 31

u uI du u du u

111 12 1 2

2 122 23 33 3

30 0

00

1 1 1

13 3 3 23

Cu 48.x

I dx

x xx

2 2 4

23 1

1

t t x2

1

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9/48

Trang 9

t

I dt

t

3 2 2

22

( 1)

2

=

t tdt t dt dt

t t

3 3 34 22

2 22 2 2

2 1 1 19 2 4 2ln

3 4 4 22 2


Cu 49. x

I x x dx

x

1

0

12 ln 1

1

Tnhx

H dx

x

1

0

1

1

. t x t tcos ; 0;

2

H 2

2

Tnh K x x dx 1

0

2 ln(1 ) . tu x

dv xdx

ln(1 )

2

K

1

2

Cu 50. I x x x dx 2

5 2 2

2

( ) 4

I =x x x dx

25 2 2

2( ) 4

=x x dx

25 2

2 4

+x x dx

22 2

2 4

= A + B.

+ Tnh A = x x dx2

5 2

2

4

. t t x . Tnh c: A = 0.

+ Tnh B = x x dx2

2 2

2

4

. t x t2sin . Tnh c: B = 2 .

Vy: I 2 .

Cu 51. x dx

I

x

2 2

41

3 4

2

Ta c:x

I dx dx

x x

2 2 2

4 41 1

3 4

2 2

.

+ Tnh I1= dx

x

2

41

3

2 = x dx

24

1

3 7

2 16

.

+ Tnhx

I dx

x

2 2

2 41

4

2

. t x t dx tdt2sin 2cos .

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10/48

Trang 10

tdt

I t dt t d t

t t

22 2 22 2

2 4 2

6 6 6

1 cos 1 1 1 3cot cot . (cot )

8 8 8 8sin sin

Vy: I 1 7 2 316

.

Cu 52.x dx

I

x

1 2

60 4

t t x dt x dx 3 23 dt

I

t

1

20

1

3 4

.

t t u u dt udu2sin , 0; 2cos2

I dt

6

0

1

3 18

.

Cu 53.x

I dxx

2

0

2

2

t x t dx tdt2cos 2sin

tI dt

22

0

4 sin 22

.

Cu 54.x dx

I

x x

1 2

20 3 2

Ta c:x dx

Ix

1 2

2 20 2 ( 1) . t x t1 2cos .

t t

I dt

t

22

22

3

(1 2cos ) 2sin

4 (2cos )

= t t dt

2

3

2

3 4cos 2cos2

=3 3

42 2

Cu 55. x x dx

1

22

0

1 2 1 t x tsin I t t tdt 6

0

3 1(cos sin )cos

12 8 8

Dng 3: Tch phn tng phn

Cu 56. I x dx 3

2

2

1

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11/48

Trang 11

tx

du dx u x

xdv dx

v x

221 1

xI x x x dx x dx

x x

3 32 2

2 22 2

3 11 . 5 2 1

2 1 1

dxx dx

x

3 32

22 2

5 2 11

I x x2 3

25 2 ln 1

I 5 2 1ln 2 1 ln22 4

Ch : Khng c dng php i bin xt

1

cos v 2;3 1;1

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12/48

Trang 12

TP3: TCH PHN HM S LNG GIC

Dng 1: Bin i lng gic

Cu 57. x xI dxx x

28cos sin2 3

sin cos

x x x

I dx x x x x dx x x

2(sin cos ) 4cos2sin cos 4(sin cos

sin cos

x x C3cos 5sin .

Cu 58.x x x

I dxx

cot tan 2 tan2

sin4

Ta c:x x x x

I dx dx dx C x x xx

2

2cot 2 2tan2 2cot 4 cos4 12

sin4 sin4 2sin4sin 4

Cu 59.

x

I dx

x x

2cos8

sin2 cos2 2

Ta c:x

I dx

x

1 cos 21 4

2 21 sin 2

4

x

dxdx

xx x

2

cos 2

1 42 2 1 sin 2 sin cos4 8 8

xdx

dx

x x2

cos 21 14

2 32 2 1 sin 2 sin4 8

x x C1 3

ln 1 sin 2 cot

4 84 2

Cu 60.dx

I

x x

3

2 3sin cos

dx

I

x3

1

21 cos

3

=dx

Ix2

3

1

42sin

2 6

=1

4 3

.

Cu 61.

I dxx

6

0

1

2sin 3

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13/48

Trang 13

Ta c: I dx dx x x

6 6

0 0

11 1 22

sin sin sin sin3 3

x x

dx dx x x

x

6 6

0 0

coscos 2 6 2 63

sin sin 2cos .sin3 2 6 2 6

x x

dx dx x x

6 6

0 0

cos sin2 6 2 61 1

2 2sin cos

2 6 2 6

x x

6 60 0

ln sin ln cos .....2 6 2 6

Cu 62. I x x x x dx 2

4 4 6 6

0

(sin cos )(sin cos )

.

Ta c: x x x x4 4 6 6(sin cos )(sin cos ) x x33 7 3

cos4 cos864 16 64

I 33

128 .

Cu 63. I x x x dx 2

4 4

0

cos2 (sin cos )

I x x dx x d x 2 22 2

0 0

1 1 1cos2 1 sin 2 1 sin 2 (sin2 ) 0

2 2 2

Cu 64. I x x dx 2

3 2

0

(cos 1)cos .

A = xdx x d x2 2 2

5 2

0 0

cos 1 sin (sin )

=8

15

B = xdx x dx2 2

2

0 0

1cos . (1 cos2 ).

2

=4

Vy I =8

15

4

.

Cu 65.2

2

0

I cos cos 2x xdx

I x xdx x xdx x x dx 2 2 22

0 0 0

1 1cos cos2 (1 cos2 ) cos2 (1 2cos2 cos4 )2 4

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14/48

Trang 14

x x x 2

0

1 1( sin2 sin4 )

4 4 8

Cu 66.

xI dxx

3

20

4sin1 cos

x x x

x x x x xx x

3 3

2

4sin 4sin (1 cos )4sin 4sin cos 4sin 2sin2

1 cos sin

I x x dx 20(4sin 2sin2 ) 2

Cu 67. I xdx 2

0

1 sin

x x x xI dx dx

22 2

0 0

sin cos sin cos2 2 2 2

x dx2

0

2 sin2 4

x xdx dx

3

22

30

2

2 sin sin2 4 2 4

4 2

Cu 68.dx

I

x

4

60cos

Ta c: I x x d x 4

2 4

0

28(1 2tan tan ) (tan )

15

.


Cu 69.xdx

Ix x

sin2

3 4sin cos2

Ta c: x xI dxx x

22sin cos

2sin 4sin 2

. t t xsin I x C

x

1ln sin 1sin 1

Cu 70.dx

I

x x3 5sin .cos

xx

dx

xxx

dxI

23233 cos.2sin8

cos.cos.sin

t t xtan . I t t t dt x x x C t x

3 3 4 2

2

3 1 3 13 tan tan 3ln tan

4 2 2tan

Ch :t

x

t2

2sin2

1

.

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15/48

Trang 15

Cu 71.dx

I

x x3sin .cos

dx dx

I

x x x x x

2 22

sin .cos .cos sin2 .cos

. t t xtan dx t

dt x

x t

2 2

2; sin2

cos 1

dt t

I dtt t

t

2

2

12

2

1

t x

t dt t C x C t

2 21 tan( ) ln ln tan

2 2

Cu 72.x x

I xdx

x

2011 2011 2009

5

sin sincot

sin

Ta c:xx

I xdx xdx

x x

2011 2011 22

4 4

11

cotsin cot cot

sin sin

t t xcot I t tdt t t C 2 4024 8046

22011 2011 20112011 2011t (1 )4024 8046

= x x C

4024 8046

2011 20112011 2011cot cot4024 8046

Cu 73.x x

I dxx

2

0

sin2 .cos

1 cos

Ta c:x x

I dxx

22

0

sin .cos2

1 cos

. t t x1 cos t

I dtt

2 2

1

( 1)2 2ln2 1

Cu 74. I x xdx 3

2

0

sin tan

Ta c:x x x

I x dx dx x x

23 32

0 0

sin (1 cos )sinsin .

cos cos

. t t xcos

u

I duu

122

1

1 3ln2

8

Cu 75. I x x dx 2

2

sin (2 1 cos2 )

Ta c: I xdx x xdx H K 2 2

2 2

2sin sin 1 cos2

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16/48

Trang 16

+ H xdx x dx 2

2 2

2sin (1 cos2 )2 2

+ K x x x xdx 2 2 2

2 2

sin 2cos 2 sin cos

xd x2

2

22 sin (sin )3

I 2

2 3

Cu 76.dx

I

x x

3

2 4

4

sin .cos

dx

I

x x

3

2 2

4

4.sin 2 .cos

. t t xtan dx

dt

x2cos

.

t dt t I t dt t

tt t

33 32 2 3

2

2 211 1

(1 ) 1 1 8 3 42 2

3 3

Cu 77.

2

2

0

sin2

2 sin

xI dx

x

Ta c:x x x

I dx dx

x x

2 2

2 20 0

sin2 sin cos2

(2 sin ) (2 sin )

.t t x2 sin .

t

I dt dt t t tt t

33 3

2 22 2 2

2 1 2 22 2 2 ln

3 22ln

2 3

Cu 78.x

I dx

x

6

0

sin

cos2

x x

I dx dx x x

6 6

20 0

sin sin

cos2 2cos 1

. t t x dt xdx cos sin

i cn: x t x t 3

0 1;6 2

Ta ct

I dt

tt

31

2

2

31 2

1 1 2 2ln

2 2 2 22 1

=

1 3 2 2ln

2 2 5 2 6

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17/48

Trang 17

Cu 79. xI e x x dx 22 sin 3

0

.sin .cos .

t t x2sin I = te t dt

1

0

1(1 )

2 = e

11

2 .

Cu 80. I x x dx 2 12sin sin

2

6

t t xcos . I 3

( 2)16

Cu 81.x

I dx

x x

4

6 60

sin4

sin cos

x

I dx

x

4

20

sin4

31 sin 2

4

. t t x23

1 sin 24 I = dtt

1

4

1

2 1

3

= t

1

1

4

4 2

3 3 .

Cu 82.

xI dx

x x

2

30

sin

sin 3cos

Ta c: x x xsin 3 cos 2cos6

;

x xsin sin 6 6

=x x

3 1

sin cos2 6 2 6

I =

x dxdx

x x

2 2

3 20 0

sin63 1

16 16cos cos

6 6

=3

6

Cu 83.x x

I dx

x

24

2

3

sin 1 cos

cos

x xI x dx x dx

x x

4 42

2 2

3 3

sin sin1 cos . sin

cos cos

x x

x dx x dx

x x

0 4

2 20

3

sin sinsin sin

cos cos

=x x

dx dx

x x

0 2 24

2 20

3

sin sin

cos cos

7

3 112

.

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18/48

Trang 18

Cu 84. I dxx x

6

0

1

sin 3 cos

I dxx x

6

0

1

sin 3 cos

= dxx

6

0

1 1

2sin

3

=x

dx

x

6

20

sin1 3

21 cos

3

.

t t x dt x dx cos sin3 3

I dt

t

1

2

20

1 1 1ln3

2 41

Cu 85. I x xdx 2

2

0

1 3sin2 2cos

I x x dx 2

0

sin 3 cos

= I x x dx x x dx 3 2

0

3

sin 3cos sin 3cos

3 3

Cu 86.xdx

I

x x

2

30

sin

(sin cos )

t x t dx dt2

tdt xdx It t x x

2 2

3 30 0

cos cos

(sin cos ) (sin cos )

dx dx

2I x

x xx

2 2 4

22 00 0

1 1cot( ) 1

2 2 4(sin cos ) sin ( )4

I 1

2

Cu 87.x x

I dx

x x

2

30

7sin 5cos

(sin cos )

Xt:

xdx xdxI I

x x x x

2 2

1 23 30 0

sin cos;

sin cos sin cos

.

t x t2

. Ta chng minh c I1 = I2

Tnh I1 + I2 =

dx dx x

x x x

2 2

220 0

1tan( ) 12

2 4sin cos 02cos ( )4

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19/48

Trang 19

I I1 21

2 I I I1 27 5 1 .

Cu 88.

x x

I dxx x

2

30

3sin 2cos

(sin cos )

t x t dx dt2

t t x x I dt dx

t t x x

2 2

3 30 0

3cos 2sin 3cos 2sin

(cos sin ) (cos sin )

x x x x

I I I dx dx dx

x x x x x x

2 2 2

3 3 20 0 0

3sin 2cos 3cos 2sin 12 1

(sin cos ) (cos sin ) (sin cos )

I

1

2 .

Cu 89.

x xI dx

x2

0

sin

1 cos

tt t t

x t dx dt I dt dt I

t t2 2

0 0

( )sin sin

1 cos 1 cos

t d tI dt I

t t

2

2 20 0

sin (cos )2

4 4 81 cos 1 cos

Cu 90.x x

I dx

x x

42

3 3

0

cos sin

cos sin

t x t dx dt2

t t x x I dt dx

t t x x

0 4 42

3 3 3 30

2

sin cos sin cos

cos sin cos sin

x x x x x x x x

I dx dx xdx

x x x x

4 4 3 32 2 2

3 3 3 30 0 0

cos sin sin cos sin cos (sin cos ) 1 12 sin2

2 2sin cos sin cos

I 1

4

.

Cu 91. I x dx x

22

20

1tan (cos )

cos (sin )

t x t dx dt2

I t dt t

22

20

1tan (sin )

cos (cos )

x dx

x

22

20

1tan (sin )

cos (cos )

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20/48

Trang 20

Do : I x x dx x x

22 2

2 20

1 12 tan (cos ) tan (sin )

cos (sin ) cos (cos )

= dt

2

0

2

I2

.

Cu 92.x x

I dx

x

4

0

cos sin

3 sin2

t u x xsin cos duI

u

2

21 4

. t u t2sin tdt

I dt

t

4 4

2

6 6

2cos

124 4sin

.

Cu 93.x

I dx

x x

3

20

sin

cos 3 sin

t t x23 sin = x24 cos . Ta c: x t2 2cos 4 vx x

dt dx

x2

sin cos

3 sin

.

I =x

dx

x x

3

20

sin.

cos 3 sin

=

x xdx

x x

3

2 20

sin .cos

cos 3 sin

=

dt

t

15

2

2

3 4

= dtt t

15

2

3

1 1 1

4 2 2

=t

t

152

3

1 2ln

4 2

=

1 15 4 3 2ln ln

4 15 4 3 2

= 1 ln 15 4 ln 3 22

.

Cu 94.x x x x

I dx

x x

2

33 2

3

( sin )sin

sin sin

x dx

I dxxx

2 2

3 32

3 31 sinsin

.

+ Tnh xI dxx

2

31 2

3 sin

. t

u x

du dxdxdv v x

x2

cotsin

I13

+ Tnhdx dx dx

I =x x

x

2 2 2

3 3 32

23 3 3

4 2 31 sin

1 cos 2cos2 4 2

Vy: I 4 2 33

.

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21/48

Trang 21

Cu 95.x

dx

x x

I2

2 20

sin2

cos 4sin

x x

dx

x

I2

20

2sin cos

3sin 1

. t u x23sin 1

udu

duu

I

2 2

1 1

22 233 3

Cu 96.x

I dxx

6

0

tan4

cos2

xx

I dx dx x x

26 6

20 0

tantan 14

cos2 (tan 1)

. t t x dt dx x dx

x

2

2

1tan (tan 1)

cos

dt

Itt

11

33

200

1 1 3

1 2( 1)

.

Cu 97.x

I dx

x x

3

6

cot

sin .sin4

x

I dxx x

3

2

6

cot2sin (1 cot )

. t x t1 cot dx dt x21sin

t

I dt t t t

3 1 3 1

3 1

3 1 3

3

1 22 2 ln 2 ln 3

3

Cu 98.dx

I

x x

3

2 4

4

sin .cos

Ta c:dx

I

x x

3

2 2

4

4.sin 2 .cos

.tdt

t x dx

t2

tan1

t dt t

I t dt t t

t t

32 2 33 3(1 ) 1 1 8 3 42( 2 ) ( 2 )2 2 3 31 1 1

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22/48

Trang 22

Cu 99.x

I dx

x x x

4

20

sin

5sin .cos 2cos

Ta c:x

I dx

x x x

4

2 20

tan 1.

5tan 2(1 tan ) cos

.t t xtan ,

t

I dt dt t tt t

1 1

20 0

1 2 1 1 2ln3 ln2

3 2 2 1 2 32 5 2

Cu 100.xdx

x x x

I

24

4 2

4

sin

cos (tan 2 tan 5)

tdt

t x dx

t2

tan1

t dt dt

I

t t t t

21 1

2 21 1

22 ln 3

32 5 2 5

Tnhdt

I

t t

1

1 21 2 5

. tt

u I du

0

1

4

1 1tan

2 2 8

. Vy I

2 32 ln

3 8

.

Cu 101.x

I dxx

22

6

sin

sin3

.

x x

I dx dx

x x x

22 2

3 2

6 6

sin sin

3sin 4sin 4cos 1

t t x dt xdx cos sin dt dt

I

tt

3

0 2

2203

2

1 1ln(2 3)

14 44 14

Cu 102.

x xI dxx

2

4

sin cos1 sin2

Ta c: x x x x x1 sin2 sin cos sin cos (v x ;4 2

)

x x

I dxx x

2

4

sin cos

sin cos

. t t x x dt x x dx sin cos (cos sin )

I dt t t

22

11

1 1ln ln2

2

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23/48

Trang 23

Cu 103.I x x xdx 2

6 3 5

1

2 1 cos .sin .cos

t

t dt

t x t x t dt x xdx dx x x

56 3 6 3 5 2

2

2

1 cos 1 cos 6 3cos sin cos sin

t tI t t dt

11 7 13

6 6

00

122 (1 ) 2

7 13 91

Cu 104.xdx

I

x x

4

20

tan

cos 1 cos

Ta c:

xdx

Ix x

4

2 20

tan

cos tan 2

.t2 2 2

2

tan

2 tan 2 tan cos

x

t x t x tdt dxx

3 3

2 2

3 2 tdt

I dtt

Cu 105.x

I dx

x x

2

30

cos2

(cos sin 3)

t t x xcos sin 3 t

I dt

t

4

32

3 1

32

.

Cu 106. xI dxx x

4

2 40

sin4

cos . tan 1

Ta c:x

I dx

x x

4

4 40

sin4

sin cos

. t t x x4 4sin cos I dt

2

2

1

2 2 2 .

Cu 107.x

I dx

x

4

20

sin4

1 cos

Ta c:x x

I dx

x

24

20

2sin2 (2cos 1)

1 cos

. t t x

2cos t

I dtt

12

1

2(2 1) 12 6ln

1 3

.

Cu 108.x

I dxx

6

0

tan( )4

cos2

Ta c:26

2

0

tan 1

(tan 1)

x

I dxx

. t t xtan

1

3

2

0

1 3

( 1) 2

dt

It

.

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24/48

Trang 24

Cu 109.36

0

tan

cos2

x

I dxx

Ta c:3 36 6tan tan

2 2 2 2cos sin cos (1 tan )0 0

x xI dx dx

x x x x

.

t t xtan

3

33 1 1 2ln

2 6 2 310

tI dt

t

.

Cu 110.x

I dx

x

2

0

cos

7 cos2

xdx

I

x

2

2 20

1 cos

2 6 22 sin

Cu 111.dx

x x

3

4 3 5

4

sin .cos

Ta c: dxx

x

x

3

384

4 3

1

sin.cos

cos

dxxx

3

24 3

4

1 1.

costan

.

t t xtan I t dt 33

84

1

4 3 1

Cu 112.3

2

0

cos cos sin( )

1 cos

x x xI x dx

x

Ta c:x x x x x

I x dx x x dx dx J K

x x

2

2 20 0 0

cos (1 cos ) sin .sin.cos .

1 cos 1 cos

+ TnhJ x x dx

0.cos .

. tu x du dx

dv xdx v x cos sin

J

2

+ Tnhx x

K dx

x2

0

.sin

1 cos

. t x t dx dt

t t t t x x K dt dt dx

t t x2 2 2

0 0 0

( ).sin( ) ( ).sin ( ).sin

1 cos ( ) 1 cos 1 cos

x x x x dx x dxK dx K

x x x2 2 2

0 0 0

( ).sin sin . sin .2

21 cos 1 cos 1 cos

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25/48

Trang 25

t t xcos dtK

t

1

21

2 1

, t t u dt u du2tan (1 tan )

u duK du u

u

2 24 4 42

44 4

(1 tan ) .2 2 2 41 tan

Vy I2

24

Cu 113.2

2

6

cosI

sin 3 cos

x

dx

x x

Ta c:2

2 2

6

sin cos

sin 3 cos

x xI dxx x

. t t x23 cos

dtIt

15

2

23

1ln( 15 4) ln( 3 2)

24


Cu 114.I x x dx 2 12sin sin .

2

6

t x t t3

cos sin , 02 2

I = tdt

42

0

3cos

2

=3 1

2 4 2

.

Cu 115.2

2 2

0

3sin 4cos

3sin 4 cos

x x

I dxx x

2 2 2

2 2 2

0 0 0

3sin 4cos 3sin 4cos

3 cos 3 cos 3 cos

x x x x

I dx dx dxx x x

2 2

2 2

0 0

3sin 4cos

3 cos 4 sin

x xdx dx

x x

+ Tnh2

1 2

0

3sin

3 cos

xI dx

x. t cos sin t x dt xdx

1

1 2

0

3

3

dt

It

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26/48

Trang 26

t 23 tan 3(1 tan ) t u dt u du 26

1 2

0

3 3(1 tan ) 3

3(1 tan ) 6

u du

Iu

+ Tnh 22 2

0

4cos4 sin

xI dxx

. t 1 1sin cos t x dt xdx1

12 12

10

4 ln34

dtI dtt

Vy:3

ln36

I

Cu 116.x

I dx

x x

4

2

6

tan

cos 1 cos

Ta c: x xI dx dx x x

x

x

4 4

2 22

26 6

tan tan1 cos tan 2cos 1

cos

t u x du dx x

2

1tan

cos

uI dx

u

1

21

3

2

. tu

t u dt du

u

2

22

2

.

I dt t

33

77 33

7 3 73 .

3 3

Cu 117.

x

I dxx x

2

4

sin4

2sin cos 3

Ta c:

x xI dx

x x

2

2

4

1 sin cos

2 sin cos 2

. t t x xsin cos I dt

t

1

20

1 1

2 2

t t u2 tan u

I du

u

1arctan22

20

1 2(1 tan ) 1 1arctan

22 22tan 2

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27/48

Trang 27


Cu 118.

x x

I dxx

3

2

3

sin

cos

.

S dng cng thc tch phn tng phn ta c:

x dxI xd J

x x x

3 33

33 3

1 4,

cos cos cos 3

vi

dxJ

x

3

3

cos

tnh J ta t t xsin . Khi

dx dt t

J x tt

3 33 2 2

2 33

23 2

1 1 2 3

ln lncos 2 1 2 31

Vy I 4 2 3

ln .3 2 3

Cu 119. xx

I e dx x

2

0

1 sin.

1 cos

Ta c:

x x

x x

x xx 2 2

1 2sin cos1 sin 12 2 tan1 cos 2

2cos 2cos2 2

x

xe dx x I e dx

x

2 2

20 0

tan2

2cos2

= e2

Cu 120.

x xI dx

x

4

20

cos2

1 sin2

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28/48

Trang 28

tu x du dx

xdv dx v

xx2

cos2 1

1 sin2(1 sin2 )

I x dx dx x x

x

4 4

20 0

1 1 1 1 1 1 1. . .4

2 1 sin2 2 1 sin2 16 2 20 cos4

x1 1 1 2 2

. tan . 0 1416 2 4 16 2 2 4 162 0

TP4: TCH PHN HM S M - LOGARIT

Dng 1: i bin s

Cu 121.x

x

eI dx

e

2

1

t x x xt e e t e dx tdt 2 2 .

tI dt

t

3

2 1

t t t t C

3 22

2 2ln 13 x x x x x e e e e e C

2

2 2ln 13

Cu 122.x

x

x x eI dx

x e

2( )

x

x

x x eI dx

x e

2( )

=

x x

x

xe x edx

xe

.( 1)

1

. t

xt x e. 1 x xI xe xe C 1 ln 1 .

Cu 123.x

dxI

e2 9

tx

t e

2

9

dt t

I Ctt2

1 3

ln6 39

x

x

e

Ce

2

2

1 9 3

ln6 9 3

Cu 124.x

x

x xI dx

ex e2

2

2 1

ln(1 ) 2011

ln ( )

Ta c:x x

I dx

x x

2

2 2

ln( 1) 2011

( 1) ln( 1) 1

. t t x2ln( 1) 1

t

I dtt

1 2010

2

t t C

11005ln

2 = x x C2 2

1 1ln( 1) 1005ln(ln( 1) 1)

2 2

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29/48

Trang 29

Cu 125.e x

x

xeJ dx

x e x1

1

( ln )

e x eex

x

d e x eJ e x

ee x 11

( ln ) 1ln ln ln

ln

Cu 126.x x

x x x

e eI dx

e e e

ln2 3 2

3 20

2 1

1

x x x x x x

x x x

e e e e e eI dx

e e e

ln2 3 2 3 2

3 20

3 2 ( 1)

1

=

x x x

x x x

e e edx

e e e

ln2 3 2

3 20

3 21

1

=x x x

e e e x 3 2 ln2 ln2ln( 1)

0 0 = ln11ln4 =

14ln

4

Cu 127.

x

dxI

e

3ln2

230

2

x

x

x

e dxI

e e

3ln2 3

20 33 2

.tx x

t e dt e dx 3 31

3 I

3 3 1ln

4 2 6

Cu 128. xI e dx ln2

3

0

1

t xe t3 1 t dt

dx

t

2

3

3

1

I = dt

t

1

3

0

13 1

1

=

dt

t

1

30

3 31

.

Tnhdt

I

t

1

1 30

31

=t

dtt t t

1

20

1 2

1 1

= ln23

Vy: I 3 ln23

Cu 129. x x

x x x x

e e dx I

e e e e

ln15 2

3ln2

24

1 5 3 1 15

t x xt e t e21 1 xe dx tdt2 .

t t dt I dt t t t t tt

4 42 4

2 33 3

(2 10 ) 3 72 2 3ln 2 7ln 2

2 24

2 3ln2 7ln6 7ln5

Cu 130.ln3 2

ln2 1 2

x

x x

e dxI

e e

t t = xe 2 xe dx tdt 2 2

I = 2 t tdt

t t

1 2

20

( 2)

1

= 2

tt dt

t t

1

20

2 11

1

= t dt

1

0

2 ( 1) +d t t

t t

1 2

20

( 1)2

1

= t t1

20( 2 ) + t t

12

02ln( 1) = 2ln3 1 .

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30/48

Trang 30

Cu 131.x x

x x

e eI dx

e e

ln3 3 2

0

2

4 3 1

t

x x x x x xt e e t e e tdt e e dx

3 2 2 3 2 3 2

4 3 4 3 2 (12 6 ) x x

tdte e dx

3 2

(2 ) 3

tdtI dt

t t

9 9

1 1

1 1 1(1 )

3 1 3 1

t t

91

1 8 ln5( ln 1) .

3 3

Cu 132. 3

16ln

3

8ln

43 dxeI x

t: x x t

t e e

2 43 4

3

tdtdx

t2

2

4

t dtI dt dt

t t

2 3 2 3 2 32

2 22 2 2

22 8

4 4

I14 3 1 8 , vi

dtI

t

2 3

1 22 4

Tnhdt

I

t

2 3

1 22 4

. t: t u u2tan , ;2 2

dt u du

22(1 tan )

I du

3

1

4

1 1

2 2 3 4 24

. Vy: I 4( 3 1)

3

Cu 133.x

x

eI dx

e

ln3

30 ( 1)

t x x xx

tdtt e t e tdt e dx dx

e

2 21 1 2 tdt

I

t

2

32

2 2 1

Cu 134.x

x

eI dx

e

ln5 2

ln2 1

t x xx

tdt t t e t e dx I t d t

e

22 32 2

11

2 201 1 2 ( 1) 2

3 3

Cu 135. xI e dx ln2

0

1

t x x xx

td td t e t e tdt e dx dx

e t

2

2

2 21 1 2

1

t

I dt dt t t

1 12

2 20 0

2 1 42 1 21 1

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31/48

Trang 31

Cu 136.x x

x xI dx

2

1

2 2

4 4 2

tx x

t 2 2

x x x x 2

4 4 2 (2 2 ) 4

1 81

ln4 ln 2 25I

Cu 137.1

0

6

9 3.6 2.4

x

x x x

dxI

Ta c:

x

x x

dx

I

1

20

3

2

3 33 2

2 2

.tx

t 3

2

.

dtI

t t

3

2

21

1

ln3 ln2 3 2

ln15 ln14

ln3 ln2

Cu 138.e

xI x x dx

x x

2

1

ln3 ln

1 ln

e e

xI dx x xdx

x x

2

1 1

ln3 ln

1 ln

=

2(2 2)

3

+

e32 1

3

=

e35 2 2 2

3

Cu 139.e

x xI dx

x

3 2

1

ln 2 ln

t t x2

2 ln x

dt dx x

2ln

I tdt

33

2

1

2 33 4 43

3 28

Cu 140.e

e

dxI

x x ex

2

ln .ln

e e

e e

dx d x I

x x x x x

2 2

(ln )

ln (1 l n ) ln (1 l n )

=

e

e

d xx x

2

1 1(ln )

ln 1 ln

= 2ln2ln3

Cu 141.x

x x

eI dx

e e

ln6 2

ln4 6 5

tx

t e . I 2 9ln3 4ln2

Cu 142.e x

I dx

x x

32

21

log

1 3ln

e e e

x

x x xdx I dx dx

xx x x x x

3

3 22

32 2 21 1 1

ln

log ln2 1 ln . ln.

ln 21 3ln 1 3ln 1 3ln

tdx

x t x t x tdtx

2 2 21 11 3ln ln ( 1) ln .3 3

.

Suy ra I t t

2

33 3

1

1 1 439ln 2 27ln 2

.

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32/48

Trang 32

Cu 143.ex x x

I dxx x

1

( 2) ln

(1 ln )

e ex

dx dx x x1 1

ln

2 (1 ln ) =e

x

e dxx x1

ln

1 2 (1 ln )

Tnh J =e

xdx

x x1

ln

(1 ln ). t t x1 ln

tJ dt

t

2

1

11 ln2

.

Vy: I e 3 2ln2 .

Cu 144.e

e

x x x xI dx

x x

3

2

2 22 ln ln 3

(1 ln )

e e

e e

I dx xdx

x x

3 3

2 2

13 2 ln

(1 ln )

e e

3 23ln2 4 2 .

Cu 145.e

x xI dx

x

22 2

21

ln ln 1

t :dx

t x dt x

ln t t t t

t t t t t I dt dt dt dt I I

e e e e

22 2 1 2

1 20 0 0 1

2 1 1 1 1

+ tt t t t

tdt dt dt dt I te

ee e e e

11 1 1 1

1 0 0 0 00

1

+t t

t t t t

tdt dt dt dt

I te te ee e e e e

2 22 2 2 2

2 1 1 1 1 21 1

1 2

Vy :e

I

e2

2( 1)

Cu 146.5

2

ln( 1 1)

1 1

x

I dxx x

t t xln 1 1 dxdtx x

21 1

I dtln3

2 2

ln2

2 ln 3 ln 2 .

Cu 147.

33

1

ln

1 ln

ex

I dxx x

tdx

t x x t tdt x

21 ln 1 ln 2 v x t3 2 3ln ( 1)

t t t t

I dt = dt t t t dt t t t

2 2 22 3 6 4 25 3

1 1 1

( 1) 3 3 1 1( 3 3 )

15ln2

4

Cu 148.e

xI dx

x x1

3 2ln

1 2ln

t t x1 2ln

e

I t dt 2

1

(2 ) =3

524

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33/48

Trang 33

Cu 149.e

x xI dx

x

3 2

1

ln 2 ln t t x

22 ln I 33 4 43 3 2

8

Cu 150.1

1

( ln )

e x

x

xeI dx

x e xt xt e xln

1ln

eeI

e.


Cu 151.

inxI e xdx

2s

0

.sin2

inxI e x xdx 2

s

0

2 .sin cos

.t x xu x du xdx

dv e xdx v esin sin

sin cos

cos

x x xI xe e xdx e e2

sin sin sin2 20 0

0

2sin .cos 2 2 2

Cu 152.

I x x x dx

1

2

0ln( 1)

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34/48

Trang 34

t

xdu dx

u x x x x

dv xdx xv

2 2

2

2 1

ln( 1) 1

2

x x xI x x dx

x x

112 3 2

2

20 0

1 2ln( 1)

2 2 1

x dxx dx dx

x x x x

1 1 1

2 20 0 0

1 1 1 2 1 3ln3 (2 1)

2 2 4 41 1

3 3ln3

4 12

Cu 153.x

I dx

x

8

3

ln

1

tu x dx

dudx xdv

v xx

ln

2 11

xI x x dx J x

883

3

12 1.ln 2 6ln8 4ln3 2

+ Tnhx

J dxx

8

3

1 . t

t tt x J tdt dt dt

t tt t

3 3 32

2 22 2 2

1 11 .2 2 2

1 11 1

tt

t

83

12 ln 2 ln3 ln2

1

T I 20ln2 6ln3 4 .

Cu 154.

exx x x

I e dx x

2

1

ln 1

e e e x

x x eI xe dx xe dx dx

x1 1 1

ln . + Tnhe ee

x x x eI xe dx xe e dx e e11

1 1

( 1)

+Tnhe e ex xe

x x ee eI e xdx e x dx e dx

x x2

11 1 1

ln ln .

Vy:e x

eI I I dx

x1 2

1

=e

e 1 .

Cu 155.e

xI x dx

x x

2

1

lnln

1 ln

Tnhe

xI dx

x x1

1

ln

1 ln

.t t x1 ln I1

4 2 2

3 3 .

+ Tnhe

I xdx 2

21

ln . Ly tch phn tng phn 2 ln c I e2 2 .

Vy I e 2 2 2

3 3

.

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35/48

Trang 35

Cu 156.2

3

2

1

ln( 1)xI dx

x

t

xdu

u x xdx

dvv

xx

2

2

32

2

ln( 1) 11

2

. Do I = x dxx x x

22

2 21

2ln( 1)

12 ( 1)

xdx

x x

2

21

ln2 ln5 1

2 8 1

dx d x

x x

2 2 2

21 1

ln2 ln5 1 ( 1)

2 8 2 1

x x2 2ln2 ln5 1ln | | ln | 1|

2 8 2 1

=

52ln2 ln5

8

Cu 157.

xI = dx

x

2

21

ln( 1)

t

dxu x du

dxxdx I xdv x x x

vxx

2

2 1

ln( 1)1 321 ln( 1) 3ln2 ln3

1 1 ( 1) 2

Cu 158.x

I x dx x

1

2

0

1ln

1

tdu dx xu x

xxdv xdx v

2

2

21ln (1 )1

2

x

I x x dx x x

1

22 2

20

11 1 2ln 2

2 1 10

xdx dx

x xx

1 1

22 2

20 0

ln3 ln3 1 ln3 1 1 21 ln

8 8 ( 1)( 1) 8 2 2 31

Cu 159.I x x dx x

22

1

1.l n

t

u xx

dv x dx

2

1ln

I 10 1

3ln3 ln23 6

Cu 160.I x x dx 1

2 2.ln(1 )

0

tu x

dv x dx

2

2

ln(1 )

I

1 4.ln2

3 9 6

Cu 161.x

I dx

x

3

21

ln

( 1)

t

u x

dxdv

x 2

ln

( 1)

I 1 3

ln3 ln4 2

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36/48

Trang 36

Cu 162.2 2

1

ln ( ln ).

1

e x x

x

x e e xI dx

e

Ta c:e e x

x

eI x dx dx H K

e

22

1 1

ln .1

+e

H x dx 2

1

ln . . t:u x

dv dx

2ln

e

H e x dx e

1

2ln . 2

+e x

x

eK dx

e

2

1 1

. t

xt e 1

ee

e

ee

t eI dt e e

t e

1

21

1 1ln

1

Vy: ee

eI e

e

1 2 ln

1

Cu 163.

2 1

1

2

1( 1 )

x

x

I x e dxx

Ta c:2 31 1

1 1

2 2

1

x x

x xI e dx x e dx H Kx

+ TnhH theo phng php tng phn I1=2 21 1 5

2

1 12 2

1 3

2

x x

x xH xe x e dx e Kx

5

23

.

2

I e

Cu 164.4

2

0

ln( 9 ) I x x dx

t u x xdv dx

2ln 9

xI x x x dx

x

4 42

20 0

ln 9 29

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37/48

Trang 37

TP5: TCH PHN T HP NHIU HM S

Cu 165.

x x

I x e dx x

31 4

2

0 1

x xI x e dx dx

x

31 1 4

2

0 01

.

+ Tnh xI x e dx 3

12

10

. t t x3 t tI e dt e e

1 1

10

0

1 1 1 1

3 3 3 3 .

+ Tnhx

I dx

x

1 4

201

. t t x4

tI dt

t

1 4

2 20

24 4

3 41

Vy:I e1 33

Cu 166. x x

I x e dx

x

2 2

31

4

xI xe dx 2

1

+x

dx

x

2 2

21

4 .

+ Tnh xI xe dx e2

21

1

+ Tnhx

I dx

x

2 2

2 21

4 . t x t2sin , t 0;

2

.

t

I dt t t

t

222

2 2

66

cos( cot )

sin

= 33

Vy: I e2 33

.

Cu 167. xxI e x x dx x

12 2 2

20

. 4 .4

x xI xe dx dx I I

x

1 1 32

1 22

0 0 4

+ Tnh x e

I xe dx

1 22

10

1

4

+ Tnhx

I dx

x

1 3

22

0 4

. t t x24 I2

163 3

3

eI

2 61

3 34 12

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38/48

Trang 38

Cu 168. xx

I e dx

x

1 2

20

1

( 1)

t t x dx dt 1 t tt t

I e dt e dt tt t

2 221 1

2 21 1

2 2 2 2

1

=e

e ee

22

1 12

Cu 169.x

x e dxI

x

23 3 1

20

.

1

t t x dx tdt 21 tI t e dt 2

2

1

( 1) t t

t e dt e J e e

22 2

1

2( )

1

+ t t t t t t t J t e dt t e te dt e e te e dt e e te e2 2 2

2 2 2 2

1 1 1

2 2 22 4 2 4 2( )

1 1 1

Vy: I e2

Cu 170.x x x

I dx

x

2 3

2

ln( 1)

1

Ta c:x x x x x x x x

f x x

x x x x

2 2 2

2 2 2 2

ln( 1) ( 1) ln( 1)( )

1 1 1 1

F x f x dx x d x xdx d x 2 2 21 1

( ) ( ) ln( 1) ( 1) ln( 1)2 2

= x x x C2 2 2 21 1 1

ln ( 1) ln( 1)4 2 2 .

Cu 171. x x x

I dx

x

4 2 3

20

ln 9 3

9

x x x x x x

I dx dx dx I I

x x x

4 4 42 3 2 3

1 22 2 2

0 0 0

ln 9 3 ln 93 3

9 9 9

+ Tnh x x

I dx

x

4 2

12

0

ln 9

9

. t x x u2ln 9 du dx

x2

1

9

u

I udu

ln9 2 2 2

1ln3

ln 9 ln 3ln9

ln32 2

+ Tnhx

I dx

x

4 3

22

0 9

. t x v2 9

xdv dx x v

x

2 2

2, 9

9

u

I u du u

5 32

23

445( 9) ( 9 )

33 3

Vy

x x xI dx I I

x

4 2 3 2 2

1 220

ln 9 3 ln 9 ln 3

3 4429

.

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39/48

Trang 39

Cu 172.e

x x xI dx

x x

3 2

1

( 1) ln 2 1

2 ln

e ex

I x dx dx x x

2

1 1

1 ln2 ln . +

ee

x ex dx

3 3

2

11

13 3

+e e

ex d x xdx x x

x x x x 1

1 1

1 ln (2 ln )ln 2 ln

2 ln 2 ln

e 2

ln2

. Vy:

e eI

3 1 2ln

3 2

.

Cu 173.e

xI dx

x x

33

1

ln

1 ln

tdx

t x x t tdt x

21 ln 1 ln 2 v x t3 2 3ln ( 1)

t t t t I dt = dt t t t dt t t t

2 2 22 3 6 4 2 5 3

1 1 1

( 1) 3 3 1 1( 3 3 ) 15 ln2

4

Cu 174.4

2

0

sin

cos

x x

I dxx

tu x du dx

xdv dx v

xx2

sin 1

coscos

x dx dx

Ix x x

4 44

0 0 0

2

cos cos 4 cos

+dx xdx

Ix x

4 4

1 20 0

cos

cos 1 sin

.t t xsin dt

I

t

22

1 20

1 2 2ln

2 2 21

Vy:2 1 2 2

ln4 2 2 2

Cu 175.4 3

2

1

ln(5 ) . 5

x x xI dx

x

Ta c:4 4

2

1 1

ln(5 )5 .

xI dx x x dx K H

x.

+x

K dx

x

4

21

ln(5 ) . t

u x

dxdv

x2

ln(5 )

K 3

ln45

+ H= x x dx 4

1

5 . . t t x5 H 164

15

Vy: I 3 164

ln45 15

Cu 176.

I x x x dx 0

22

(2 ) ln(4 )

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40/48

Trang 40

Ta c: I x x dx 2

0

(2 ) + x dx2

2

0

ln(4 ) = I I1 2

+ I x x dx x dx

2 22

10 0

(2 ) 1 ( 1) 2

(s dng i bin: x t1 sin )

+x

I x dx x x dx

x

2 2 222 2

2 0 20 0

ln(4 ) ln(4 ) 24

(s dng tch phn tng phn)

6ln2 4 (i bin x t2tan )

Vy: I I I1 23

4 6 ln22

Cu 177.8 ln

13

xI dx

x

tu x dx

dudx x

dvv xx

ln

2 11

xI x x dx

x

88

33

12 1ln 2

+ Tnhx

J dxx

8

3

1 . t t x 1

t dtJ dt

t t

3 32

2 22 2

2 12 1 2 ln3 ln2

1 1

I 6ln8 4ln3 2(2 ln3 ln2) 20ln2 6ln3 4

Cu 178. dxxx

xI

2

1

3

2

ln1

Ta c: I xdx xx

2

31

1 1ln

.t

u x

dv dx xx

3

ln1 1

( )

I x x x dx xx x

22

4 511

1 1 1ln ln ln

4 4

=

21 63 1ln2 ln 264 4 2

Cu 179.e

xx x xI e dx

x

2

1

ln 1

Ta c:

e e e xx x e

I xe dx e xdx dx H K J x1 1 1ln

+e e

x x e x eH xe dx xe e dx e e1

1 1

( 1)

+e e ex xe

x x e ee eK e xdx e x dx e dx e J

x x11 1 1

ln ln

Vy: e e e eI H K J e e e J J e1 1 .

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41/48

Trang 41

Cu 180.x x

I dx

x

2

3

4

cos

sin

Ta cx

x x2 3

1 2cos

sin sin

. t

u x

xdv dx

x3

cos

sin

du dx

v

x2

1

2sin

I = xx

2

2

4

1 1.

2 sin

+dx

x

x

2 2

2

44

1 1 1( ) cot

2 2 2 2 2sin

=

1

2.

Cu 181.

x x

I dxx

4

30

sin

cos

t:u x du dx

xdv dx v

x x3 2

sin 1

cos 2.cos

x dxI x

x x

44 4

2 200 0

1 1 1tan

2 4 2 4 22cos cos

Cu 182. dxx

xxI

2

0

2

2sin1

)sin(

Ta c:x x

I dx dx H K x x

22 2

0 0

sin1 sin2 1 sin2

+x x

H dx dx x

x

2 2

20 01 sin2

2cos4

. t:

u xdu dx

dxdv

v xx

21

tan2cos 2 4

4

x

H x x

22

0 0

1tan ln cos

2 4 2 4 4

+x

K dxx

22

0

sin

1 sin2

. t t x2

xK dx

x

22

0

cos

1 sin2

dx

K x

x

2 2

20 0

12 tan 1

2 42cos

4

K 1

2

Vy, I H K 1

4 2

.

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42/48

Trang 42

Cu 183.x x x x

I dx

x

3

20

(cos cos sin )

1 cos

Ta c:

x x x x x

I x dx x x dx dx J K x x

2

2 20 0 0

cos (1 cos ) sin .sin

.cos .1 cos 1 cos

+ Tnh J x x dx 0

.cos .

. tu x

dv xdx cos

J x x x dx x 0 0

0

( .sin ) sin . 0 cos 2

+ Tnhx x

K dx

x2

0

.sin

1 cos

. t x t dx dt

t t t t x x K dt dt dx

t t x2 2 2

0 0 0

( ).sin( ) ( ).sin ( ).sin

1 cos ( ) 1 cos 1 cos

x x x x dx x dxK dx K

x x x2 2 2

0 0 0

( ).sin sin . sin .2

21 cos 1 cos 1 cos

t t x dt x dx cos sin . dtK

t

1

21

2 1

, t t u dt u du2tan (1 tan )

u duK du u

u

2 24 44

2

44 4

(1 tan ).

2 2 2 41 tan

Vy I2

24

Cu 184.x x x x

I dx

x x

2

32

3

( sin )sin

(1 sin )sin

Ta c:x x x x dx

I dx dx H K xx x x

2 2 223 3 3

2 2

3 3 3

(1 sin ) sin

1 sin(1 sin )sin sin

+x

H dx

x

2

3

23

sin

.t

u xdu dx

dx

dv v xx

2 cotsin

H3

+dx dx dx

Kx x

x

2 2 2

3 3 3

23 3 3

3 21 sin

1 cos 2cos2 4 2

Vy I 3 23

Cu 185.x x

I dxx

23

0

sin

1 cos2

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43/48

Trang 43

Ta c:x x x x

I dx dx dx H K x x x

2 23 3 3

0 0 2 0 2

sin sin

1 cos2 2cos 2cos

+x x

H dx dx x x

3 30 2 0 2

1

22cos cos

.tu x

du dxdxdv v x

x2

tancos

H x x xdx x 3 3300 0

1 1 1tan tan ln cos ln2

2 2 22 3 2 3

+x

K dx xdx

x

223 3

0 2 0

sin 1tan

22cos

x x 301 1

tan 32 2 3

Vy:

I H K 1 1 3 1 1

ln2 3 ( 3 ln2)2 2 3 6 22 3

Cu 186.I x x dx 3

0

1sin 1.

t t x 1 I t t tdt t tdt x xdx 2 2 2

2 2

1 1 1

.sin .2 2 sin 2 sin

t du xdxu xv xdv xdx

2 42cossin

I x x x xdx

222

11

2 cos 4 cos

t u x du dx

dv xdx v x

4 4

cos sin

. T suy ra kt qu.

Cu 187. xx

I e dx x

2

0

1 sin.

1 cos

x

xe dx x I e dx

x x

2 2

20 0

1 sin

2 1 coscos

2

+ Tnh x x

x x

xI e dx e dx xx

2 2

120 0

2sin .cossin 2 2

1 cos2cos

2

xxe dx

2

0

tan2

+ Tnhx

e dxI

x

2

220

1

2cos

2

. t

xxu e

du e dxdx

dv xvx2 tan2cos 2

2

xx

I e e dx

22

20

tan2

Do : I I I e21 2

.

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44/48

Trang 44

Cu 188.x

xI dx

e x

20

cos

(1 sin2 )

x

xI dx

e x x

20 2

cos

(sin cos )

. tx x

x x x dx

u due edx x

dv vx xx x

2

cos (sin cos )

sin

sin cos(sin cos )

x x x

x x xdx xdxI

x xe e e

2 22

0 0 0

cos sin sin sin.sin cos

t

x x

u x du xdx

dxdv v

e e

1 1

1 1

sin cos

1

x x x

xdx xdxI x

e e ee

2 22

0 0 02

1 cos 1 cossin .

tx x

u x du xdx

dxdv v

e e

2 2

1 1

cos sin

1

x x

xdxI x I I e

e ee e

222

0 02 2

1 1 sin 1cos . 1 2 1

e

I2 1

2 2

Cu 189. x

x xI dx

6 64

4

sin cos6 1

t t x dt dx t xt x

t t x x I dt dx

6 6 6 64 4

4 4

sin cos sin cos6 6

6 1 6 1

xx

x xI dx x x dx

6 64 46 6

4 4

sin cos2 (6 1) (sin cos )

6 1

x dx4

4

5 3cos4

8 8

5

16

I 5

32

.

Cu 190.x

xdxI

46

6

sin

2 1

Ta c:

x x x

x x x

xdx xdx xdxI I I

04 4 46 6

1 20

6 6

2 sin 2 sin 2 sin

2 1 2 1 2 1

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45/48

Trang 45

+ Tnhx

x

xdxI

0 4

1

6

2 sin

2 1

. t x t t

t t x

t t xI dt dt dx

0 0 04 4 4

1

6 6 6

2 sin ( ) sin sin

2 1 2 1 2 1

x

x x

xdx xdxI xdx x dx

4 46 6 6 64 2

0 0 0 0

sin 2 sin 1sin (1 cos2 )

42 1 2 1

x x dx

6

0

1(3 4cos2 cos4 )

8

4 7 3

64

Cu 191.e

I x dx

1

cos(ln )

tt t

t x x e dx e dt ln

tI e tdt

0

cos

= e1

( 1)2

(dng pp tch phn tng phn).

Cu 192. xI e x xdx 2

2sin 3

0

.sin .cos

t t x2sin t

I e t dt e

1

0

1 1(1 )

2 2 (dng tch phn tng phn)

Cu 193.I x dx 4

0

ln(1 tan )

t t x4

I t dt

4

0

ln 1 tan4

=

tdt

t

4

0

1 tanln 1

1 tan

= dt

t

4

0

2ln

1 tan

= dt t dt 4 4

0 0

ln2 ln(1 tan )

= t I40.ln2

I2 ln24

I ln2

8

.

Cu 194.I x x dx 2

0

sin ln(1 sin )

tx

u x du dx x

dv xdx v x

1 cosln(1 sin )

1 sinsincos

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46/48

Trang 46

x x

I x x x dx dx x dx x x

22 2 2

0 0 0

cos 1 sincos .ln(1 sin ) cos . 0 (1 sin ) 12

1 sin 1 sin 20

Cu 195.x x

I dxx

4

0

tan .ln(cos )

cos

t t xcos dt xdx sin t t

I dt dt

t t

1

12

2 211

2

ln ln .

t

u t

dv dt t2

ln

1

du dt t

vt

1

1

I

2

2 1 ln22

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47/48

Trang 47

TP6: TCH PHN HM S C BIT

Cu 196.Cho hm sf(x) lin tc trn R v f x f x x 4( ) ( ) cos vi mixR.

Tnh: I f x dx 2

2

( )

.

t x = t f x dx f t dt f t dt f x dx 2 2 2 2

2 2 2 2

( ) ( )( ) ( ) ( )

f x dx f x f x dx xdx 2 2 2 4

2 2 2

2 ( ) ( ) ( ) cos

I 3

16

Ch : x x x4 3 1 1

cos cos2 cos48 2 8

.

Cu 197.Cho hm sf(x) lin tc trn R v f x f x x ( ) ( ) 2 2cos2 , vi mixR.

Tnh: I f x dx

3

2

3

2

( )

.

Ta c : I f x dx f x dx f x dx

3 302 2

03 3

2 2

( ) ( ) ( )

(1)

+ Tnh : I f x dx 0

1

32

( )

. t x t dx dt I f t dt f x dx 3 3

2 2

10 0

( ) ( )

Thay vo (1) ta c: I f x f x dx x x dx 3 3 3

2 2 2

0 0 0

( ) ( ) 2 1 cos2 2 cos

xdx xdx

32 2

0

2

2 cos cos

x x20

3

22 sin sin 6

2

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48/48

Cu 198.x

I dx

x x

4

2

4

sin

1

I x xdx x xdx I I 4 4

21 2

4 4

1 sin sin

+ Tnh I x xdx 4

21

4

1 sin

. S dng cch tnh tch phn ca hm s l, ta tnh c I1 0 .

+ Tnh I x xdx 4

2

4

sin

. Dngpp tch phn tng phn, ta tnh c: I22

24

Suy ra: I 2

24

.

Cu 199.

5

2

3 2 1

1 1

x

x

e x xI dx

e x x

5 5 5 5

2 2 2 2

3 2 1 1 1 2 1 2 1

1 1 1 1 1 1

x x x x

x x x

e x x e x x e x e xI dx dx dx dx

e x x e x x e x x

5 5

2 2

5 2 1 2 13

2 1( 1 1) 1( 1 1)

x x

x x

e x e xx dx dx

x e x x e x

t 2 1

1 12 1

x

x e x

t e x dt dxx

5

2

52 1 5

22

1

2 12 2 13 3 2ln 3 2ln

11

e

e

e eI dt I t

t ee

Cu 200.

xI dxx x x

24

20( sin cos )

.

x x x

I dxx x x x

4

20

cos.

cos ( sin cos )

. t

xu

x

x xdv dx

x x x2

coscos

( sin cos )

x x xdu dx

x

vx x x

2

cos sin

cos1

sin cos

x dx

I dx

44

=4

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200 Bai Tap Tich Phan Megabook