Bai Tap Tich Phan Day Du LTDH

7/22/2019 Bai Tap Tich Phan Day Du LTDH

1/108

THY NGUYN QUANG SN.T 0909 230 970

1

TCH PHN

LUYN THI I HC

THY NGUYN QUANG SN

0909 230 970108/53b,Trn Vn Quang,F10,Tn Bnh.


2/108


3/108


3

Ch :

Phn tch

1

0

3 xI dx

1 x

, ri t t 1 x s tnh nhanh hn.2. i bin s dng 2

Cho hm s f(x) lin tc trn on [a;b], tnh ( )

b

a

f x dx ta thc hin cc bc sau:

Bc 1.t x = u(t) v tnh / ( )dx u t dt .Bc 2.i cn: ,x a t x b t .

Bc 3. /( ) [ ( )] ( ) ( )b

a

f x dx f u t u t dt g t dt

.

V d 1. Tnh tch phn

1

2

20

1I dx

1 x

.

Giit x sin t, t ; dx cos tdt

2 2

I CN :1

x 0 t 0, x t2 6

6 6

20 0

cos t cos tI dt dt

cost1 sin t

6

60

0

dt t 06 6

. Vy I 6

.

V d 2. Tnh tch phn

2

2

0

I 4 x dx .

Hng dn:t x 2 sin t S: I .

V d 3. Tnh tch phn

1

2

0

dxI

1 x

.

Gii

t 2x tan t, t ; dx (tan x 1)dt2 2

x 0 t 0, x 1 t4

4 42

2

0 0

tan t 1I dt dt

41 tan t

. Vy I 4

.

V d 4. Tnh tch phn

3 1

2

0

dxI

x 2x 2

.

Hng dn:


4/108


4

3 1 3 1

2 2

0 0

dx dxI

x 2x 2 1 (x 1)

. t x 1 tan t ; S: I 12

.

V d 5. Tnh tch phn

2

20

dxI

4 x

. S: I 2

.

V d 6. Tnh tch phn3 1

2

0

dxI

x 2x 2

. S: I 12

.

3. Cc dng c bit3.1. Dng lng gic

V d 11 (bc sin l). Tnh tch phn2

2 3

0

I cos x sin xdx

.

Hng dn:

t t cos x S:2

I

15

.

V d 12 (bc cosin l). Tnh tch phn2

5

0

I cos xdx

.

Hng dn:

t t sin x S:8

I15

.

V d 13 (bc sin v cosin chn). Tnh tch phn2

4 2

0

I cos x sin xdx

.

Gii2 2

4 2 2 2

0 0

1I cos x sin xdx cos x sin 2xdx

4

2 2

2

0 0

1 1(1 cos 4x)dx cos2x sin 2xdx

16 4

2 2

2

0 0

1 1(1 cos 4x)dx sin 2xd(sin2x)

16 8

3 2

0

x 1 sin 2xsin 4x

16 64 24 32

.

Vy I 32

.

V d 14. Tnh tch phn2

0

dxI

cos x sin x 1

.

Hng dn:

tx

t tan2

. S: I ln 2 .


5/108


5

Biu din cc hm s LG theo tan2

at :

2

2 2 2

2 1 2sin ; cos ; tan .

1 1 1

t t ta a a

t t t

3.2. Dng lin kt

V d 15. Tnh tch phn

0

xdxI

sin x 1

.

Giit x t dx dt .I CN x 0 t , x t 0

0

0

( t)dt tI dt

sin( t) 1 sin t 1 sin t 1

0 0

dt dtI I

sin t 1 2 sin t 1

22

0 0

dt dttt t2 4 cossin cos2 42 2

2 00

td

2 4 ttan2 t 2 2 4

cos2 4

. Vy I .

Tng qut:

0 0

xf(sin x)dx f(sin x)dx2

.

V d 16. Tnh tch phn2 2007

2007 2007

0

sin xI dxsin x cos x

.Gii

t x t dx dt2

.I CN: x 0 t , x t 0

2 2

20070

2007 2007

2

sin t2I dx

sin t cos t2 2

2 2007

2007 2007

0

cos tdx J

sin t cos t

(1).

Mt khc2

0

I J dx2

(2). T (1) v (2) suy ra I

4

.

Tng qut:

2 2n n

n n n n

0 0

sin x cos xdx dx , n

sin x cos x sin x cos x 4

.


6/108


7/108


7

Tng qut:Vi a > 0 , 0 , hm s f(x) chn v lin tc trn on ; th

x

0

f(x)dx f(x)dx

a 1

.

V d 20. Cho hm s f(x) lin tc trn

v tha f( x) 2f(x) cos x .

Tnh tch phn2

2

I f(x)dx

.

Gii

t2

2

J f( x)dx

, x t dx dt .C: x t , x t2 2 2 2

2 2

2 2

I f( t)dt J 3I J 2I f( x) 2f(x) dx

2 2

02

cos xdx 2 cos xdx 2

. Vy2

I3

.

3.3. Cc kt qu cn nh

i/ Vi a > 0 , hm s f(x) l v lin tc trn on [a; a] th

a

a

f(x)dx 0

.

ii/ Vi a > 0 , hm s f(x) chn v lin tc trn on [a; a] th

a a

a 0f(x)dx 2 f(x)dx

.

iii/ Cng thc Walliss (dng cho trc nghim)

2 2

n n

0 0

(n 1)!!,

n !!cos xdx sin xdx(n 1)!!

. ,n !! 2

neu n le

neu n chan

.

Trong n!! c l n wallissv c nh ngha da vo n l hay chn. Chng hn:

0!! 1; 1 !! 1; 2 ! ! 2; 3 !! 1.3; 4 ! ! 2.4; 5 !! 1.3.5; 6!! 2.4.6; 7 !! 1.3.5.7; 8 !! 2.4.6.8; 9!! 1.3.5.7.9; 10 !! 2.4.6.8.10 .

V d 21.2

11

0

10!! 2.4.6.8.10 256cos xdx

11!! 1.3.5.7.9.11 693

.

V d 22.2

10

0

9!! 1.3.5.7.9 63sin xdx . .

10!! 2 2.4.6.8.10 2 512

.


8/108


8

II. TCH PHN TNG PHN1. Cng thcCho hai hm s u(x), v(x) lin tc v c o hm trn on [a; b]. Ta c

/ / / // /uv u v uv uv dx u vdx uv dx

b b b

a a ad uv vdu udv d(uv) vdu udv b b b b

b ba a

a a a a

uv vdu udv udv uv vdu .

Cng thc:b b

ba

a a

udv uv vdu (1).

Cng thc (1) cn c vit di dng:b b

b/ /a

a a

f(x)g (x)dx f(x)g(x) f (x)g(x)dx

(2).

2. Phng php gii ton

Gi s cn tnh tch phn

b

a

f(x)g(x)dx ta thc hinCch 1.Bc 1.t u f(x), dv g(x)dx (hoc ngc li) sao cho d tm nguyn hm v(x) v vi

phn /du u (x)dx khng qu phc tp. Hn na, tch phnb

a

vdu phi tnh c.Bc 2. Thay vo cng thc (1) tnh kt qu.

c bit:i/ Nu gp

b b b

ax

a a a

P(x)sin axdx, P(x)cosaxdx, e .P(x)dx Vi P(x) l a thc th t u P(x) .

ii/ Nu gp

b

a

P(x) ln xdx th t u ln x .Cch 2.

Vit li tch phn

b b

/

a a

f(x)g(x)dx f(x)G (x)dx v s dng trc tip cng thc (2).

V d 1. Tnh tch phn1

x

0

I xe dx .

Gii

t x x

u x du dx

dv e dx v e

1 1

11x x x x0 0

0 0

xe dx xe e dx (x 1)e 1 .


9/108


9

V d 2. Tnh tch phn

e

1

I x ln xdx .

Gii

t 2

dxduu ln x

xdv xdx xv

2

e ee2 2

11 1

x 1 e 1x ln xdx ln x xdx2 2 4 .

V d 3. Tnh tch phn2

x

0

I e sin xdx

.

Gii

t x x

u sin x du cos xdx

dv e dx v e

2 2

x x x2 20

0 0

I e sin xdx e sin x e cos xdx e J

.

t x x

u cos x du sin xdx

dv e dx v e

2 2

x x x20

0 0

J e cos xdx e cos x e sin xdx 1 I

22

e 1I e ( 1 I) I

2

.

Ch :i khi ta phi i bin s trc khi ly tch phn tng phn.

V d 7. Tnh tch phn

2

4

0

I cos xdx

.

Hng dn:

t t x2

0

I 2 t cos tdt 2

.

V d 8. Tnh tch phn

e

1I sin(ln x)dx . S:

(sin1 cos1)e 1I 2

.

III. TCH PHN CHA GI TR TUYT I:Phng php gii ton:1. Dng 1:


10/108


10

Gi s cn tnh tch phn

b

a

I f(x) dx , ta thc hin cc bc sauBc 1. Lp bng xt du (BXD) ca hm s f(x) trn on [a; b], gi s f(x) c BXD:

x a 1x 2x b f(x) 0 0

Bc 2. Tnh1 2

1 2

b x x b

a a x x

I f(x) dx f(x)dx f(x)dx f(x)dx .

V d 9. Tnh tch phn

2

2

3

I x 3x 2 dx

.

GiiBng xt du

x 3 1 2 2x 3x 2 0 0

1 2

2 2

3 1

59I x 3x 2 dx x 3x 2 dx

2

. Vy59

I2

.

V d 10. Tnh tch phn2

2

0

I 5 4 cos x 4 sin xdx

. S: I 2 3 2 6

.

2. Dng 2

Gi s cn tnh tch phn b

a

I f(x) g(x) dx , ta thc hin

Cch 1.

Tch b b b

a a a

I f(x) g(x) dx f(x) dx g(x) dx ri s dng dng 1 trn.Cch 2.Bc 1. Lp bng xt du chung ca hm s f(x) v g(x) trn on [a; b].Bc 2. Da vo bng xt du ta b gi tr tuyt i ca f(x) v g(x).

V d 11. Tnh tch phn 2

1

I x x 1 dx

.

Gii

Cch 1. 2 2 2

1 1 1

I x x 1 dx x dx x 1 dx

0 2 1 2

1 0 1 1

xdx xdx (x 1)dx (x 1)dx


11/108


11

0 2 1 22 2 2 2

1 0 1 1

x x x xx x 0

2 2 2 2

.

Cch 2.Bng xt du

x 1 0 1 2

x 0 + +x 1 0 +

0 1 2

1 0 1

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

120 21 10x x x x 0 . Vy I 0 .3. Dng 3

tnh cc tch phn b

a

I max f(x), g(x) dx v b

a

J min f(x), g(x) dx , ta thchin cc bc sau:

Bc 1. Lp bng xt du hm s h(x) f(x) g(x) trn on [a; b].Bc 2.+ Nu h(x) 0 th max f(x), g(x) f(x) v min f(x), g(x) g(x) .+ Nu h(x) 0 th max f(x), g(x) g(x) v min f(x), g(x) f(x) .

V d 12. Tnh tch phn 4

2

0

I max x 1, 4x 2 dx .

Giit 2 2h(x) x 1 4x 2 x 4x 3 .

Bng xt dux 0 1 3 4

h(x) + 0 0 +

1 3 4

2 2

0 1 3

80I x 1 dx 4x 2 dx x 1 dx

3 .

Vy80

I3

.

V d 13. Tnh tch phn

2

x

0

I min 3 , 4 x dx .

Gii

t x xh(x) 3 4 x 3 x 4 .Bng xt du

x 0 1 2h(x) 0 +

1 2 21x 2x

0 10 1

3 x 2 5I 3 dx 4 x dx 4x

ln 3 2 ln 3 2

.Vy2 5

Iln 3 2

.


12/108


12

IV. BT NG THC TCH PHNPhng php gii ton1. Dng 1

chng minh

b

a

f(x)dx 0 (hocb

a

f(x)dx 0 ) ta chng minh f(x) 0 (hoc f(x) 0

) vi x a; b .

V d 14. Chng minh

1

3 6

0

1 x dx 0 .

Gii

Vi

1

3 36 6 6

0

x 0; 1 : x 1 1 x 0 1 x dx 0 .2. Dng 2

chng minh

b b

a a

f(x)dx g(x)dx ta chng minh f(x) g(x) vi x a; b .

V d 15. Chng minh2 2

10 11

0 0

dx dx

1 sin x 1 sin x

.

Gii

Vi 11 10x 0; : 0 sin x 1 0 sin x sin x2

10 1110 11

1 11 sin x 1 sin x 0

1 sin x 1 sin x

.

Vy2 2

10 110 0

dx dx

1 sin x 1 sin x

.

3. Dng 3

chng minh

b

a

A f(x)dx B ta thc hin cc bc sau.Bc 1. Tm gi tr ln nht v nh nht ca f(x) trn on [a; b] ta c m f(x) M .

Bc 2. Ly tch phn

b

a

A m(b a) f(x)dx M(b a) B .

V d 16. Chng minh

1

2

0

2 4 x dx 5

.

Gii

Vi 2 2x 0; 1 : 4 4 x 5 2 4 x 5 .

Vy

1

2

0

2 4 x dx 5 .


13/108


13

V d 17. Chng minh

3

4

2

4

dx

4 23 2 sin x

.

Gii

Vi 23 2 1x ; : sin x 1 sin x 14 4 2 2

22

1 11 3 2 sin x 2 1

2 3 2 sin x

3

4

2

4

1 3 dx 31

2 4 4 4 43 2 sin x

. Vy3

4

2

4

dx

4 23 2 sin x

.

V d 18. Chng minh3

4

3 cotx 1dx

12 x 3

.

Gii

Xt hm scotx

f(x) , x ;x 4 3

ta c

2/

2

xcotx

sin xf (x) 0 x ;4 3x

f f(x) f x ;3 4 4 3

3 cotx 4

x ;x 4 3

3

4

3 cotx 4dx

3 4 x 3 4

. Vy3

4

3 cotx 1dx

12 x 3

.

4. Dng 4 (tham kho)

chng minh

b

a

A f(x)dx B (m dng 3 khng lm c) ta thc hin

Bc 1. Tm hm s g(x) sao cho

b

b

a

a

f(x) g(x) x a; b

f(x)dx B

g(x)dx B

.

Bc 2. Tm hm s h(x) sao cho

b

b

a

a

h(x) f(x) x a; b

A f(x)dxh(x)dx A

.


14/108


14

V d 19. Chng minh

2

2

20070

2 dx

2 41 x

.Gii

Vi

2007 22 1

x 0; : 0 x x2 2

2 2007

2007 2

1 1 11 x 1 x 1 1

2 1 x 1 x

2 2 2

2 2 2

2007 20 0 0

dx dxdx

1 x 1 x

.

t x sin t dx cos tdt .C:2

x 0 t 0, x t2 4

2

2 4

20 0

dx cos tdt

cos t 41 x

. Vy

2

2

20070

2 dx

2 41 x

.

V d 20. Chng minh

1

20

3 1 xdx 2 1

4 2x 2 1

.

Gii

Vi 2x 0; 1 : 2 1 x 2 1 3 1

2

x x x

3 1 2 1x 2 1

1 1 1

2

0 0 0

xdx xdx xdx

3 1 2 1x 2 1

.

Vy

1

20

3 1 xdx 2 1

4 2x 2 1

.

V. NG DNG CA TCH PHN

A. TNH DIN TCH HNH PHNG:1. Din tch hnh thang congCho hm s f(x) lin tc trn on [a; b]. Din tch hnh thang cong gii hn bi cc ng

y f(x), x a, x b v trc honh lb

a

S f(x) dx

.

Phng php gii tonBc 1. Lp bng xt du hm s f(x) trn on [a; b].

Bc 2. Da vo bng xt du tnh tch phn

b

a

f(x) dx .

V d 1. Tnh din tch hnh phng gii hn bi y ln x, x 1, x e v Ox.Gii


15/108


15

Do ln x 0 x 1; e nn

e e

e1

1 1

S ln x dx ln xdx x ln x 1 1 . Vy S 1 (vdt).

V d 2. Tnh din tch hnh phng gii hn bi 2y x 4x 3, x 0, x 3 v Ox.

GiiBng xt dux 0 1 3y 0 + 0

1 3

2 2

0 1

S x 4x 3 dx x 4x 3 dx 1 33 3

2 2

0 1

x x 82x 3x 2x 3x

3 3 3

. Vy

8S

3 (vdt).

2. Din tch hnh phng

2.1. Trng hp 1.Cho hai hm s f(x) v g(x) lin tc trn on [a; b]. Din tch hnh phng gii hn bi cc ng

y f(x), y g(x), x a, x b lb

a

S f(x) g(x) dx .Phng php gii tonBc 1. Lp bng xt du hm s f(x) g(x) trn on [a; b].

Bc 2. Da vo bng xt du tnh tch phn

b

a

f(x) g(x) dx .2.2. Trng hp 2.Cho hai hm s f(x) v g(x) lin tc trn on [a; b]. Din tch hnh phng gii hn bi cc ng

y f(x), y g(x) l S f(x) g(x) dx

. Trong , l nghim nh nht v lnnht ca phng trnh f(x) g(x) a b .Phng php gii tonBc 1. Gii phng trnh f(x) g(x) .Bc 2. Lp bng xt du hm s f(x) g(x) trn on ; .

Bc 3. Da vo bng xt du tnh tch phn f(x) g(x) dx

.

V d 3. Tnh din tch hnh phng gii hn bi cc ng 3 2y x 11x 6, y 6x ,x 0, x 2 .

Giit 3 2 3 2h(x) (x 11x 6) 6x x 6x 11x 6

h(x) 0 x 1 x 2 x 3 (loi).

Bng xt du:


16/108


17/108


17

1 33 32 2

0 1

x x 162 2x 3x 2x 3x

3 3 3

. Vy

16S

3 (vdt).

V d 7. Tnh din tch hnh phng gii hn bi 2y x 4x 3 v y x 3 .Gii

Phng trnh honh giao im2x 4x 3 x 3

2

2

x 3 0x 0

x 4x 3 x 3x 5

x 4x 3 x 3

.

Bng xt dux 0 1 3 5

2x 4x 3 + 0 0 +

1 3 5

2 2 2

0 1 3

S x 5x dx x 3x 6 dx x 5x dx

1 3 53 2 3 2 3 2

0 1 3

x 5x x 3x x 5x 1096x

3 2 3 2 3 2 6

. Vy

109S

6 (vdt).

V d 8. Tnh din tch hnh phng gii hn bi 2y x 1 , y x 5 .Gii

Phng trnh honh giao im2 2x 1 x 5 t 1 t 5, t x 0

2

2

t x 0t x 0

t 1 t 5 x 3

t 3t 1 t 5

3 3

2 2

3 0

S x 1 x 5 dx 2 x 1 x 5 dx

Bng xt du

X 0 1 32x 1 0 +

1 3

2 2

0 1

S 2 x x 4 dx x x 6 dx 1 33 2 3 2

0 1

x x x x 732 4x 6x3 2 3 2 3 . Vy 73S 3 (vdt).

Ch :Nu hnh phng c gii hn t 3 ng tr ln th v hnh (tuy nhin thi H th khng c).

B. TNH TH TCH KHI TRN XOAY:1. Trng hp 1.


18/108


18

Th tch khi trn xoay do hnh phng gii hn bi cc ng y f(x) 0 x a;b ,

y 0 , x a v x b (a b) quay quanh trc Ox lb

2

a

V f (x)dx .

V d 9. Tnh th tch hnh cu do hnh trn 2 2 2(C) : x y R quay quanh Ox.

GiiHonh giao im ca (C) v Ox l 2 2x R x R .Phng trnh 2 2 2 2 2 2(C) : x y R y R x

R R

2 2 2 2

R 0

V R x dx 2 R x dx

R3 3

2

0

x 4 R2 R x

3 3

. Vy

34 RV

3

(vtt).

2. Trng hp 2.Th tch khi trn xoay do hnh phng gii hn bi cc ng x g(y) 0 y c;d ,

x 0 , y c v y d (c d) quay quanh trc Oy ld

2

c

V g (y)dy .

V d 10. Tnh th tch hnh khi do ellipse2 2

2 2

x y(E) : 1

a b quay quanh Oy.

Gii

Tung giao im ca (E) v Oy l2

2

y1 y b

b .

Phng trnh2 2 2 2

2 22 2 2

x y a y(E) : 1 x a

a b b

b b2 2 2 2

2 22 2

b 0a y a yV a dy 2 a dyb b

R2 3 22

20

a y 4 a b2 a y

33b

. Vy

24 a bV

3

(vtt).

3. Trng hp 3.Th tch khi trn xoay do hnh phng gii hn bi cc ng y f(x), y g(x) , x a v

x b (a b, f(x) 0, g(x) 0 x a; b ) quay quanh trc Ox lb

2 2

a

V f (x) g (x) dx .

V d 11. Tnh th tch hnh khi do hnh phng gii hn bi cc ng2

y x ,2

y x quayquanh Ox.Gii

Honh giao im4

x 0 x 0

x 1x x

.


19/108


19

1 1

4 4

0 0

V x x dx x x dx

1

5 2

0

1 1 3x x

5 2 10

. Vy

3V

10

(vtt).

4. Trng hp 4.Th tch khi trn xoay do hnh phng gii hn bi cc ng x f(y), x g(y) , y c vy d (c d, f(y) 0, g(y) 0 y c; d ) quay quanh trc Oy l

d

2 2

c

V f (y) g (y) dy .

V d 12. Tnh th tch hnh khi do hnh phng gii hn bi cc ng 2x y 5 ,x 3 y quay quanh Oy.

Gii

Tung giao im 2y 1

y 5 3 yy 2

.

2

2 22

1

V y 5 3 y dy

2

4 2

1

y 11y 6y 16 dy

25 3

2

1

y 11y 1533y 16y

5 3 5

. Vy

153V

5

(vtt).

PHNG PHAP TNH TCH PHAN

I. PHNG PHP I BIN S:

Du hiu Cch chn

2 2a x t x = |a| sint; vi ;

2 2t

hoc x = |a| cost; vi 0;t

2 2x a

t x =a

sint; vi ; \ 0

2 2t

hoc x =a

cost; vi

0; \

2t

2 2a x t x = |a|tant; vi ;

2 2t

hoc x = |a|cost; vi 0;t

a x

a x

hoc

a x

a x

t x = acos2t


20/108


20

a b x t x = a + (b a)sin2t

2 2

1

a x t x = atant; vi ;

2 2t

Bi 1: Tnh1 2

2

2

2

1 xI dxx

Gii:

t x = cost, ;2 2

t

. dx = - sint d.

i cn:

x 2

2

4

t 1 0

Khi :1 2

2

2

2

1 xI dx

x

=

0 2

2

4

1 os .c t sint dt

cos t

=

4

20

sin .sint tdt

cos t

=24

20

sin tdt

cos t

=

4

20

11 dt

cos t

= tan 4

0t t

= 14

. (v 0;

4t

nn sint 0 sin sint t )

Bi 2: Tnh 2 2 2

0

a

I x a x dx

Gii:t x = asint, ;

2 2t

. dx = acostdt

i cn:X 0 A

T 02

Khi : 2 2 2

0

a

I x a x dx = 2

2 2 2 2

0

sin 1 sin .a t a t acostdt

=2

4 2 2

0

sina tcos tdt

=

4 22

0

sin 24

atdt

= 4 2

0

1 48

acos t dt

=4 1 sin4 2

8 40

at t

=4

16a

Bi 3: Tnh1

2 2

0

1I x x dx

Gii:


21/108


21

t x = sint, ;2 2

t

. dx = costdt.

i cn:x 0 1

t

0 2

Khi :1

2 2

0

1I x x dx =2

2 2

0

sin 1 sin .t t costdt

=2

2 2

0

1sin

4tcos tdt

=2

2

0

1sin 2

4tdt

=

= 2

0

11 4

8cos t dt

=1 1

sin4 28 4

0t t

=16

Bi 4: Tnh1

3 2

0

1I x x dx

Gii: t t = 21 x t2 = 1 x2 xdx = -tdti cn:

x 0 1t 1 0

Khi :1

3 2

0

1I x x dx =1

2 2

0

1I x x xdx = 1

2

0

1 . .t t tdt = 1

2 4

0

t t dt =

3 5 1

03 5

t t

=

2.

15

Bi 5: Tnh

2

5ln

e

e

dx

I x

Gii: t t = lnx dt =dx

i cn:x e e2t 1 2

Khi :

2

5ln

e

e

dxI

x =

2

51

dt

t=

4

21 15.

14 64t

Bi 6: Tnh 1

43 4

0

1I x x dx

Gii: t t = x4 + 1 dt = 4x3dx 34

dtx dx

i cn:x 0 1t 1 2


22/108


22

Khi : 1

43 4

0

1I x x dx =2

4 5

1

21 1 31.

14 20 20t dt t

Bi 7: Tnh2

5

0

sinI xcoxdx

Gii: t t = sinx ; dt cosxdx i cn:

x 02

t 0 1

Khi :12

5 5

0 0

1sin

6I xcoxdx t dt

.

Bi 8: Tnh

124

0 tanI xdx

Gii:

Ta c:12 12

0 0

sin4tan4

4xdx dx

cos x

t t = cos4x ; 4s 4 sin 44

dtdt in xdx xdx

i cn:

x 012

t 1 12

Khi :

1112 12 2

10 0 12

1sin 4 1 1 1 1

tan 4 ln ln 2.14 4 4 4 4

2

x dt dtI xdx dx t

cos x t t

Bi 9: Tnh2

5

0

I cos xdx

Gii:

Ta c: 2 2 2

25 4 2

0 0 0

1 sincos xdx cos xcoxdx x coxdx

t t = sinx ; dt cosxdx i cn:

x 02

t 0 1


23/108


24/108


24

Bi 13: Tnh2

2sin

0

sin2xI e xdx

Gii:t t = sin2x ; s 2dt in xdx

i cn:

x 0 2

t 0 1

Khi :2

12sin

0 0

1sin 2 1.

0x t tI e xdx e dt e e

Bi 14: Tnh2

20

sin2

1

xI dx

cos x

Gii:t t = 1 + cos2x ; s 2 s 2dt in xdx in xdx dt i cn:

x 0 2

t 2 1

Khi : 1 22

20 2 1

2sin2ln ln 2.

11

x dt dtI dx t

cos x t t

Bi 15: Tnh

43

0 tanI xdx

Gii:

t t = tanx ; 2 2 21 tan 1 1dt

dt x dx t dt dxt

i cn:

x 0 4

t 0 1Khi :

21 1 1 1 13 24

32 2 2 2

0 0 0 0 0 0

2

111 2 1tan 01 1 2 1 2 2 1

11 1 1 1 1ln 1 ln 2 1 ln 2 .

02 2 2 2 2

d tt t t t I xdx dt t dt tdt dtt t t t

t

Bi 16: Tnh1

0

1

1I dx


25/108


25

Gii:

t t = x ; 2 2t x dx tdt i cn:

x 0 1t 0 1

Khi : 1 1 1

0 0 0

11 12 2 1 2 ln 1 2 1 ln 2 .01 11

tI dx dt dt t tt tx

Bi 17: Tnh1

33 4

0

1I x x dx

Gii:

t t = 3 4 3 4 3 23

1 14

x t x x dx t dt

i cn:x 0 1t 1 0

Khi :1 1

33 4 3 4

0 0

13 3 31 .

04 16 16I x x dx t dt t

Bi 18: Tnh0

21

1

2 4I dx

x x

Gii:

Ta c:

0 0

22 21 1

1 1

2 4 1 3dx dx

x x x

t 1 3 tan t vi

2; . 3 1 tan2 2

t dx t dt

i cn:

x -1 0

t 06

Khi :0 6

21 0

1 3 3 3.6

2 4 3 3 180

I dx dt tx x

Bi 19: Tnh

1 3

80 1

xI dxx

Gii:

Ta c:

1 13 3

28 40 01 1

x xdx dx

x x

t 4 tan t vi 3 21

; . 1 tan2 2 4

t x dx t dt


26/108


26

i cn:x 0 0

t 04

Khi :

1 13 3 24 4

28 240 0 0 0

1 1 tan 1 1 .41 4 1 tan 4 4 161 0x x tI dx dx dt dt t

x tx

Bi 20: Tnh1

1 lne xI dx

x

Gii:

t 21 ln 1 ln 2dx

t x t x tdt x

i cn:x 1 et 1 2

Khi : 2 2 32

1 1 1

2 2 2 11 ln 2.2 2 2 .

3 31

ex t

I dx t tdt t dtx

Bi 21: Tnh 1

0

ln 2

2

xI dx

x

Gii:

t ln 22

dxt x dt

i cn:x 1 1t ln2 0

Khi : 1 0 ln 2 2 2

0 ln 2 0

ln 2ln 2 ln 2.

02 2 2

x tI dx tdt tdt

x

Bi 22: Tnh2

20 1 sin

cosxI dx

x

Gii:

t sin tanx t vi 2; 1 tan2 2

t cosxdx t dt

i cn:x 0

2

t 04


27/108


27

Khi :22 4 4

2 20 0 0

1 tan

1 sin 1 tan 4

cosx t I dx dt dt

x t

Bi 23: Tnh2

3

1

sinI dx

Gii:

t 22

1 2tan 1 tan

2 2 2 1

x dtt dt dx dx

t

Ta tnh:2

2

1 1 2 1.

2sin 11

tdtdx dt

t t t

t

i cn:

x

3

2

t 3

3 1

Khi : 12

33 3

11 1 3 1

ln ln ln 3.3sin 3 23

I dx dt tx t

Bi 24: Tnh 1

1

1 ln

e

I dxx x

Gii:

t 1 ln dxt x dt

i cn:x 1 et 1 2

Khi :

2

1 1

21ln ln 2.

11 ln

edt

I dx tx x t

Bi 25: Tnh3

15

0

xI x e dx

Gii:

t 3 2 233

dtt x dt x dx x dx

i cn:x 0 1t 0 1


28/108


29/108


29

2 2 3 32

23 3 31 1 2 2

2

2 1 1 1

3 1 3 1 11 1

3 31 1 1 1 1 2 1 1 2 1 1 1ln 1 ln 1 ln ln ln ln ln

3 3 1 3 2 3 32 12 2 2 2 1 2 1

dx x dx dt I dt

t t tx x x x

tt t

t

Bi 28: Tnh2 3

20

3

2 1

xI dx

x x

Gii:

Ta c:

2 23 3

220 0

3 3

2 1 1

x xdx dx

x x x

t 1t x dt dx i cn:

x 0 2t 2 3

Khi :

3 3 22 2 3 33 3

22 2 20 0 1 1

3 22 2 2

1

3 3 3 13 13 3

2 1 1

39 1 33 9 3 3 9 9 ln 3 3 1 9 3 1 9 ln 3 ln1 1 3 9ln

12 2

t t ttx xI dx dx dt dt

x x t tx

tt t dt t t

t t

Bi 29: Tnhln 2 2

20

33 2

x x

x x

e eI dxe e

Gii:t x xt e dt e dx i cn:

x 0 ln2t 1 2

Khi :

ln 2 ln 2 2 22

2 2 20 0 1 1

2 2

1 1

3 3 3 2 1

3 2 3 2 3 2 1 2

2 21 1 3 4 9 42 2 ln 1 ln 2 2 ln 3 ln 2 ln 4 ln 3 2ln ln ln ln l1 11 2 2 3 4 3

x x xx

x x x x

e e e t I dx e dx dt dt

e e e e t t t t

dt dt t t t t

Bi 30: Tnh

4

1 1

dxI

x x


30/108


30

Gii:t 2 2t dx tdt i cn:

x 1 4t 1 2

Khi :

4 2 2 2

21 1 1 1

2 1 12 2

1 1 11

2 2 1 42 ln ln 1 2 ln ln 2ln .

1 3 2 3

dx tdt dt I dt

t t t t t t x x

t t

Bi 31: Tnh 1

32

0

1I x dx

Gii:t sin , 0;2

t t dx costdt

i cn:x 0 1

t 02

Khi :

21 2 2 2 23 32 2 3 4

0 0 0 0 0

2 2 2 2 22 2

0 0 0 0 0

2

0

1 21 1 sin . .

2

1 1 1 1 1 1 sin 2 11 2 2 2 2 2 2 . . 1 42

4 4 2 8 4 2 2 2 80

1 1

8 8 8

cos tI x dx t costdt cos t costdt cos tdt dt

tcos t cos t dt dt cos tdt cos tdt cos

dt co

2

0

1 sin 4 34 . .2

8 16 8 4 8 16 160

ts tdt

Bi 32: Tnh2

3

6

I cos xdx

Gii:


31/108


31

32 2 2 2

3 2 2 2

6 6 6 6

sin 2. 1 sin 1 sin sin sin3

61 1 1 5

13 2 24 24

xI cos xdx cos x cosxdx x cosxdx x d x x

Bi 33: Tnh4

4 40

sin4

sin

xI dx

x cos x

Gii:

4 4 4 4

4 4 4 4 2 220 0 0 0

42 2

20

sin 4 2sin 2 2 2sin 2 2 2sin 2 2

1sin sin 1 2sin 1 sin 22

1 1 1 11 sin 2 ln 1 sin 2 ln ln 24

1 2 2 21 sin 2 02

x xcos x xcos x xcos xI dx dx dx dx

x cos x x cos x xcos x x

d x x

x

Bi 34: Tnh32

4

1 sin

cos xI dx

Gii:

23 22 2 2 2

4 4 4 4

2 2 2

4 4 4

1 sin1 sin

1 sin 1 sin 1 sin

1 1 3 2 22sin s 2 sin sin 22 4 4

4

xcos x cos xI dx cosxdx cosxdx x cosxdx

x x x

cosx cosx x dx cosxdx in xdx x x

Bi 35: Tnh2

4

sin

sin

x cosxI dx

x cosx

Gii:

2 2

4 4

sinsin 2ln sin ln 2sin sin

4

d x cosxx cosxI dx x cosx

x cosx x cosx


32/108


32

Bi 36: Tnh2

3

0

sinI xdx

Gii:

32 2 2

3 2 2

0 0 0

1 2sin sin sin 1 12

3 3 30

cos xI xdx x xdx cos x d cosx cosx

Bi 37: Tnh3

sin

cos xI dx

x

Gii:

2 23 2

0

2

4 3 4 1 sin 33 4 3. . sin

sin sin sin sin

1 14sin sin 4. sin ln sinsin1 2

cos x xcos x cos x cosxI dx dx cosxdx d x

x x x x

x d x x x C

Bi 38: Tnhs 3

sin

in xI dx

3

2s 3 3s 4sin 3 4sin 3 2 1 2sin sin

13 2 2. sin 2 sin 2

2

in x inx xI dx dx x dx x cos x dx

x x

x x x c x x C

Bi 39: Tnh1

4 20 1

xI dx

x x

Gii: t 2 2t x dt xdx

i cn:x 0 1t 0 1

Khi :1 1

24 20 0

1

1 2 1 3

2 4

x dtI dx

x xt

t 12

y t dy dt

i cn:t 0 1

y1

2

3

2


33/108


33

Khi :

31 2

2 210 22

1 1

2 21 3 32 4 4

dt dyI

t y

t 3 24 3

z y dz dy

i cn:

y1

2

3

2

z1

3 3

Khi :

33 32

2 221 1 12

2 3 3

1 3 13 32 4 133

4 44

dy dz dz I

zz

y

t 2tan 1 tanu dz u du i cn:

z1

3 3

u6

3

Ta c:3 23

2 21

63

1 1 1 tan 1 3

1 1 tan3 3 3 6 36

dz uI du u

z u

Bi 40: Tnh

1

20 2 1

xI dx

x

Gii:

t 12 12 2

t dtt x x dx

i cn:x 0 1t 1 3

Khi :

1 3 3

2 2 20 1 1

131 1 1 1 1 1 22 . ln ln 312 4 4 4 32 1

tx dt

I dx dt tt t t t x

Bi 41: Tnh 0

92

1

1I x x dx

Gii:


34/108


34

t 1t x dt dx i cn:

x -1 0t 0 1

Khi :

0 1 1 1

9 22 9 2 9 11 10 9

1 0 0 0

12 11 10

1 1 2 1 2

1 1 2 1 12

012 11 10 12 11 10 660

I x x dx t t dt t t t dt t t t dt

t t t

Bi 42: Tnh2

0 1

dxI

cosx

Gii:

2 2 2

2 20 0 0

2tan 12

1 22 02 2

xd

dx dx xI

x xcosx cos cos

Bi 43: Tnh1

15 8

0

. 1 3 .I x x dx

Ta c:1 1

15 8 8 8 7

0 0

. 1 3 . . 1 3 .x dx x x x dx

t 8 71 3 2424

dtt x dt x dx dx

i cn:x 0 1

t 1 4Khi :

5 31 1 4 4 2 23 115 8 8 8 7 2 2

0 0 1 1

41 1 1 1 29. 1 3 . . 1 3 . . .

5 3 13 24 72 72 2702 2

t t tI x x dx x x x dx t dt t t dt

Bi 44: Tnh1 3

20 1

xI dx

x x

Gii:


35/108


35

3 2 3 21 1 1 13

3 2 4

2 22 2 20 0 0 0

1 1 1 153 2 4 2 2 2 2

0 0 0 0

1 11

11 1 1

1 11 1. 1.

05 5J

x x x x x xxI dx dx dx x x x dx

x xx x x x x x

xx x dx x dx x x xdx x x xdx

t 2 1 2t x dt xdx i cn:

x 0 1t 1 2

Khi :

2 2 2 2

3 3 5 31 12 2 2 2 2 2

1 1 1 1

5 3

2 2

2 21 1 1 1 1 21 .

1 12 2 2 2 5 3

2 1 2 1 4 2 2 2 2 2 2 25 5 3 3 5 3 15 15 15

J t t dt t t dt t dt t dt t t

Vy2 2 1

15 15I

Bi 45: Tnh4

20

sin4

1

xI dx

cos x

Gii:

Ta c:4 4

2 20 0

sin 4 2sin 2 2

1 1

xcos xdx dx

cos x cos x

t 21 2sin sin 2t cos x dt xcosxdx xdx 2 21 2 2 1 2 1 1 2 3cos x t cos x cos x t t

i cn:

x 04

t 23

2

Khi :

3 322 2

32 22

22 2 3 6 6

4 4 4 6ln 32

3 3 44 2 6 ln 2 ln 2 6 ln

2 2 3

t dt

I dt dt t tt t t


36/108


36

Bi 46: Tnh2

4

1 sin 2

dxI

x

Gii:

2 2 2 2

2 22

4 4 4 4

1 1 12tan1 sin 2 2 2 4 2sin

2 4 44

dx dx dx dxI x

x x cosx cos xcos x

Bi 47: Tnh

4

30

s 2

sin 2

co xI dx

x cosx

Gii:

Ta c:

4 4

3 30 0

sin sins 2sin 2 sin 2

cosx x cosx xco x dx dxx cosx x cosx

t sin 2 sint cosx x dt cosx x dx x 0

4

t 2 2 2 Khi :

2 2 2 2

3 2 3 20 0

2 1 2 1 1 1 1 1 12 2

3 92 2 6 4 20

1 2 2 2 2 1 2 2 1 4 2 4 9 4 2 5

9 9 96 4 2 2 3 2 2 2 2 1 18 2 1 18 2 1

tI dt dt

t t t t t

Bi 48: Tnh4

0

s 2

sin 2

co xI dx

x cosx

Gii:

Ta c: 4 4

0 0

sin sins 2

sin 2 sin 2

cosx x cosx xco xdx dx

x cosx x cosx

t sin 2 sint cosx x dt cosx x dx i cn:

x 04

t 2 2 2 Khi :


37/108


37

2 2 2 2

0 0

2 2 2 21 2ln 2 2 2ln 2 2 3 2ln 3

0

32 1 2 ln 3 ln 2 2 2 1 2 ln

2 2

tI dt dt t t

t t

Bi 49: Tnh 2

32

0

sin 2 1 sinI x x dx

Gii:

t 21 sin 2 2sin sin 2t x dt xcosxdx xdx i cn:

x 02

t 1 2

Khi :

2 4232 3

0 1

2 1 15

sin 2 1 sin 414 4 4

t

I x x dx t dt

Bi 50: Tnh 2

2

0

sin 1I xcosx cosx dx

Gii:Ta c:

2 2 2

2 2 2 3

0 0 0

sin 1 sin 1 2 2 .sinI xcosx cosx dx xcosx cosx cos x dx cosx cos x cos x xdx

t sint cosx dt xdx i cn:

x 02

t 1 0

Khi : 0 1 2 3 4

2 3 2 3

1 0

12 172 2

02 3 4 12

t t tI t t t dt t t t dt

Bi 51: Tnh2

2 2 2 20

sin

sin

xcosxI dx

a cos x b x

Gii: Ta c:

2 2 2

2 2 2 2 2 2 2 2 2 2 2 20 0 0

sin sin sin

sin 1 sin sin sin

xcosx xcosx xcosxI dx dx dx

a cos x b x a x b x b a x a


38/108


38

t

2 22 2 2 2 2 2 2 2 2

2 2

2 2 sinsin sin

sin

tdt b a xcosxdx

t b a x a t b a x a tdtxcosxdx

b a

i cn:x 0

2

t |a| |b|

Khi : 2 22 22 2

1 1.

b

a

b b atdtI t

b a a bt b a a b a

Bi 52: Tnh2

30

1

3 2

xI dx

x

Gii:

t 33 23 23 2 3 2 3 3 ;3

tt x t x t dt dx x

i cn:x 0 2t 3 2 2

Khi :

3 3

3

2 2 5 22 4

32 2

221 1 1 42 4 2 37 4 23 . 1

3 3 5 2 3 5 5 152

tt t

I t dt t t dtt

Bi 53: Tnh4

27 9

dxIx x

Gii:

t 2 2 2 2 29 9 0 ; 9dx tdt tdt

t x t x t tdt xdxx x t

i cn:

x 7 4t 4 5

Khi :5

24

51 3 1 7ln ln

49 6 3 6 4

dt t

t t

Bi 54: Tnh4

0 1 tan

dxI

Gii:

t 22 2 21tan 1 tan 1 tan 1dt dt

t x dt dx x dx dxcos x x t


39/108


39

i cn:

x 04

t 0 1Khi :

1 2 3

1 1

2 22 20 0

1 1 11 1 1 1 12 1 2 1 2 1 2 11 1 2 1 0 0 0

J J J

dt t dt tdt dt I dtt t t t t t t

Tnh: 110

11 1 ln 2ln 1

02 1 2 2

dtJ t

t

Tnh: 21 1

22 2 2

0 0

1 11 1 1 ln 2ln 1

02 1 4 1 4 4

d ttdtJ t

t t

Tnh: 1 43 20 0

1 12 1 2 8

dtJ dut

(vi t = tanu)

Vyln 2 ln 2 ln 2

2 4 8 8 4I

Bi 55: Tnh2

3

sin

dxI

Gii:

Ta c:2 2 2

2 2

3 3 3

sin sinsin sin 1 sdx xdx xdx

x co x

t sint cosx dt xdx i cn:

x3

2

t1

2 0

Khi :

1 1 1 10 2 2 2 2

2 21 0 0 0 02

11 1 1 1 1 1 ln 1 ln 1 2

1 1 2 1 1 2 1 2 1 20

dt dt dt dt I dt t tt t t t t t

1 1 3ln ln

2 2 2

1 1 1ln ln 3

2 3 2

Bi 56: Tnh1

20

sinx xI dx

cos x


40/108


40

Gii:

Ta c:

1 2

1 1 1

2 2 20 0 0

sin sin

I I

x xdx xI dx dx

cos x cos x cos x

Tnh 31 20

dxIcos x

t2

1tan

u xdu dx

v xdv dxcos x

p dng cng thc tnh tch phn tng phn ta c:

3 3 3 3

1 20 0 0 0

3 sin 3 3tan tan ln3 3

3 3 30 0

3 1ln3 2

d cosxxdx xI x x xdx dx cosx

cos x cosx cosx

Tnh 3 32 2 20 0

sin 12 1 13

0

d cosxxI dx

cos x cos x cosx

Vy3

ln 2 13

I

Bi 57: Tnh1 3

20 1

xI dx

x x

Gii:Ta c:

3 2 3 21 1 1 13

3 2 42 22 2 2

0 0 0 0

1 1 1 153 2 4 2 2 2 2

0 0 0 0

1 11

11 1 1

1 11. 1. 1.

05 5

x x x x x xxI dx dx dx x x x dx

x xx x x x x x

xx x dx x x x xdx x x xdx

t 2 1 2t x dt xdx i cn:x 0 1t 1 2


41/108


41

Khi :

2 2 2 2

3 31 12 2 2 2

1 1 1 1

5 32 25 3

2 2

1 1 1 1 1 1 11 .

2 5 2 5 5 2 2

21 1 2 1 2 1 2 2 1 1 1 2 4 2 2 2 1 2 2. .

15 2 5 2 3 5 5 5 5 3 3 5 5 3 15 15

I t t dt t t dt t dt t dt

t t

Bi 58: Tnh1

1 5 4

xI dx

x

Gii: t 5 4 4t x dt dx

i cn:X -1 1T 9 1

Khi :

1 1 9 9 9

1 9 1 1 1

3

5 11 5 5 1 14 4

16 8 165 4 2

9 95 1 2 5 1 5 13 1. 3 1 27 1

1 18 16 3 8 24 4 12 6

tdt

x t

I dx dt dt tdtx t t t

t t

Bi 59: Tnh9

3

1

1I x xdx

Gii: t 1t x dt dx

i cn:

X 1 9T 0 -8Khi :

9 8 0

74 4 73 43 3 3 3 3

1 0 8

03 3 3 3 4681 1 2 2

84 7 4 7 7I x xdx t t dt t t dt t t

Bi 60: Tnh3

6sin sin

6

dxI

x x

Gii:

3 3 3

2

6 6 6

2

3 sin sin3 1sin sin sin sin6 2 2

dx dx dxI

xcosxx x x x cosx


42/108


42

3 3 3

2 2

6 6 6

3

6

2 tan tan22 3

s 3 tan tan tan 3 tan 1 3 tan 3 tan 1

1 12 3 tan3 tan 3 tan 1

d x d xdx

co x x x x x x x

d xx x

3 3

6 6

3 tan 1tan 3 32 2 2 ln tan 2 ln 3 tan 1tan 3 tan 1

6 6

1 32 ln 3 ln 2 ln 4 ln 2 2ln 3 2 ln 2 ln

23

d xd xx x

x x

Bi 61: Tnh1

20 3

x

dxI

e

Gii:

t x xt e dt e dx i cn:

x 0 1t 1 e

Khi :

21

2 2 2 2 2 2 2 2

0 1 1 1 1

22 2 2

2 21

1 2 1

3 2 23 3 3 31 1 1 1 1 1 3

. ln ln 3 2 ln12 3 3 6 6 4

e e e e

x

e

d tdx dt tdt tdt I

e t t t t t t t t

e ed t t t

t t

Bi 62: Tnh

1

22 11 5

dxI

Gii: t 11 5 5t x dt dx

i cn:

x -2 1t 1 6

Khi :

1 6

2 22 1

61 1 1 1 1

15 5 30 5 611 5

dx dt I

t tx

Bi 63: Tnh

1

sin lne xI dx

x

Gii:


43/108


43

t ln dxt x dt i cn:

x 1 et 0 1

Khi : 1

1 0

1sin ln sin 1 0 1 10

e

xI dx tdt cost cos cos cosx

Bi 64: Tnh5

2

3

9I x dx

Gii:

t2

2

2 2 22

2

99

2

9 9 99

2 2 2

tt x x x

t

t t tt x t dx dt

t t t

i cn:x 3 5t 3 9

Khi :5 9 92 2 2

22 3 2

3 3 3

99 9 9 81 9 819 . ln ...

32 2 4 2 4 8 2 6

t t t t I x dx dt dt t

t t t t t

Bi 65: Tnh

4

2

12

1

sinI dx

x cosx

Gii:

4 4

22

12 12

1 1 1 1 34cot2 2 4 2sin sin

4 12

I dx dx xx cosx x

Bi 66: Tnh1

0

sinI xdx

t 2t x dx tdt i cn:

x 0 1

t 0 1Khi :1

0

2 sinI t tdt t sinu t du dt

dv tdt v cosx

p dng cng thc tnh tch phn tng phn ta c:

1

0

1 1 12 2 2 2 sin 2 sin1 1

0 0 0I tcost costdt tcost t cos


44/108


44

II. PHNG PHP TCH PHN TNG PHN

Tch phn cc hm s dng P(x)sinax; P(x)cosax; P(x)eaxtrong P(x) l mt a thc t

...

u P x

dv

Tch phn cc hm s dng P(x)lnx trong P(x) l mt a thc t ln...

u x

dv

Bi 1: Tnh1

2

0

xI xe dx

t22 1

2xx

du dxu x

v edv e dx

p dng cng thc tnh tch phn tng phn:

1 1 1 2

2 2 2 2 2 2 2 2 2

0 0 0

1 11 1 1 1 1 1 1 1 12 1

0 02 2 2 4 2 4 2 4 4x x x x x e

I xe dx xe e dx e e d x e e e e

Bi 2: Tnh3

20

xI dx

cos x

t2

tan

co

u xdu dx

dxv xdv

s x


3 3 342

0 0 0 0

3 sin 3tan tan3

3 30

d cosxx xI dx x x xdx dx

cos x cosx cosx

3 3ln ln 23

3 30

cosx

Bi 3: Tnh1

2

0

xI x e dx

t2 2

xx

du xdxu x

v edv e dx



45/108


45

1 1 12 2

0 0 0

12 2

0x x x x

I x e dx x e xe dx e xe dx

Tip tc tnh:1

0

xJ xe dx t x x

u x du dx

dv e dx v e

p dng cng thc tnh tch phn tng phn:1 1

0 0

11

0x x xJ xe dx xe xe dx

Vy I = e - 2

Bi 4: Tnh 1

3

0

3 1 xI x e dx

t33

33 11

3xx

du dxu x

v edv e dx


1 1 1

3 3 3 3 3 3

0 0 0

1 11 1 13 1 3 1 3 1

0 03 3 3x x x x x xI x e dx x e e dx x e e d e

3 3 31 11 1 2 5

3 10 03 3 3

x xx e e

e

Bi 5: Tnh2

2

0

sinI x xdx

Ta c:

2 2 2 22

0 0 0 0

1 2 1sin 22 2

cos xI x xdx x dx xdx xcos xdx

2 220

22 8

0

xxdx

Tnh 20

2cos xdx

t 12 sin22

du dxu x

dv cos xdx v x


2 2

0 0

1 1 2 12 sin 2 sin 2 02 2

2 2 4 20 0

cos xxcos xdx x x xdx


46/108


46

Vy22

2

0

4sin

16I x xdx

Bi 6: Tnh2

sin

0

sin2xI e xdx

Gii:

Ta c:2 2

sin sin

0 0

sin 2 2 sinx xI e xdx e xcosxdx

t sint x dt cosxdx i cn:

x 02

t 0 1

Khi :12

sin

0 0

2 sin 2x tI e xcosxdx te dt

tt t

u t du dt

dv e dt v e


0 0

1 1 11

0 0 0t t t t t te dt te e dt te e Vy I = 2

Bi 7: Tnh 1

4 1 ln

e

I x xdx

t 2

ln

4 12

dxu x du

xdv x dx

v x x


2 2 2 21 1

4 1 ln 2 ln 2 1 2 21 1

e ee eI x xdx x x x x dx e e x x e

Bi 8: Tnh 1

2

0

ln 1I x x dx

t 2 1 2t x dt xdx i cn:

x 0 1t 1 2

Khi :


47/108


47

1 2

2

0 1

1ln 1 ln

2I x x dx tdt t

lndx

u t dut

dv dt v t


1 1

2ln ln 2ln 2 11tdt t t dt . Vy

1

2

0

1ln 1 ln 2 2I x x dx

Bi 9: Tnh 2

6

ln sinI cosx x dx

t ln sin

sinos sin

cosxu x du dx

dv c dxv x


2 2

6 6

12 2 2ln sin sin ln sin in ln sin sin ln 2 12

6 6 6

I cosx x dx x x cosxdx x x x

Bi 10: Tnh3

2

4

sin

dxI

x

t2

cotsin

u xdu dx

dx v xdvx


3 32

4 4

9 4 31 1 33 3cot cot . ln sin lnsin 3 36 2 23

4 4

xdxI x x xdx x

x

Bi 11: Tnh2

0

cosxI e xdx

tos sin

x x

u c x du xdx

dv e dx v e


1

2 2

0 0

cos sin20

x x x

I

I e xdx e cosx e xdx


48/108


48

Tnh2

1

0

sinxI e xdx

. tsin

x x

u x du cosxdx

dv e dx v e

p dng cng thc tnh tch phn tng phn

2 2

1

0 0

sin sin s sin2 20 0

x x x xI e xdx e x e co xdx e x I

Suy ra:2 2

0

1 1cos sin2 2

2 20 0

x x x eI e xdx e cosx e x

Bi 12: Tnh2

0

1 sin.

1 osxxxI e dx

c

Ta c:

2

1

2 2 2 2 2

20 0 0 0 0

1 sin sin 1 sin. . .1 osx 1 osx 1 osx 2 1 osxcos

2

x x

x x x

I

I

x e dx x e dx xI e dx e dx e dxxc c c c

Tnh:2

120

1

2 cos2

xe dxI

t2 tan 2

2

xxu e du e dx

dxxdv

vxcos

p dng cng thc tnh tch phn tng phn

2 2 22

120 0 0

1tan tan . tan .2

2 2 2 2cos 02

xx x xe dx x x x

I e e dx e e dxx

Tnh:2 2 2

220 0 0

2sin cosin 2 2. . tan .1 osx 22

2

x x x

x xs

x xI e dx e dx e dx

xccos

Vy 2I e

TNH TCH PHN BNG CCH GIN TIP


49/108


49

Bi 1: Tnh2

0

sin

sin

xI dx

x cosx

Gii: t2

x t dx dt

i cn:

x 0 2

t2

0

Khi :

0 2 2

0 02

sin2 s s

s s int s s insin

2 2

tco t co x

I dt dt dxco t co x x

t cos t

Vy2 2

0 0

sin2 2

2 4sin 0

x cosxI I I dx dx x I

x cosx

Bi 2: Tnh32

3 30

sin

sin

xI dx

cos x

Gii:

t2

t dx dt

i cn:

x 0 2

t2

0

Khi :

30 3 32 2

3 3 3 33 3 0 0

2

sins s2

s sin s sinsin s2 2

tco t co x

I dt dt dxco t t co x x

t co t

Vy3 32 2

3 30 0

sin2 2

sin 2 40

x cos xI I I dx dx x I

x cos x

Bi 3: Tnh cc tch phn:1

0

x

x x

eI dx

e e

v

1

0

x

x x

eI dx

e e


50/108


50

Ta c:1

0

1I J dx

1 1 21

0 0

1 1ln ln ln 2 ln

0 2

x xx xx x

x x x x

d e ee e eI J dx e e e e

e e e e e

T suy ra:21 1

1 ln2 2

eI

e

v

2

1 21 ln

2 1

eJ

e

Bi 4: Tnh2

0

1 s inxln

1+cosxI dx

Gii:t2

x t dx dt

i cn:

x 0

2

t2

0

Khi :

0 2 2

0 02

1 s in1 s t 1 s x2

ln ln ln1+sint 1+sinx1+cos

2

tco co

I dt dt dx

t

Vy

2 2 2 2

0 0 0 0

1 1 s 1 1 s2 ln ln ln . ln1 0 01 s inx 1 s x 1 s inx 1 s x

cosx inx cosx inxI I I dx dx dx dx Ico co

Bi 5: Tnh62

6 60

sin

sin

xI dx

cos x

Gii:

t2

t dx dt

i cn:

x 0 2

t2

0

Khi :


51/108


51

60 6 62 2

6 6 6 66 6 0 0

2

sins s2

s sin s sinsin s2 2

tco t co x


t co t

Vy6 62 2

6 60 0

sin2 2sin 2 4

0

x cos xI I I dx dx x I

x cos x

Bi 6: Tnh 2

0

sin sinI x nx dx

Gii: t t t dt dx i cn:

x 0 2 t

Khi :

sin sin sin sinI t n t dt t n nt dt

sin sin sin sin st nt cos n dt t nt in n dt

sin sinI t nt cos n dt

(do sin 0n )

t t dy dt i cn:

t

y Khi :

sin sin sin sin sin sinI y ny cos n dy y ny cos n dy y ny cos n d

sin sin t nt cos n dy I

0I I I

Bi 7: Tnh

2

30

4sin

sin

xI dx

x cosx

Gii: t2

x t dx dt

i cn:

x 0 2

t2

0

Khi :


52/108


52

0 2 2

3 3 30 0

2

4sin4 s 4 s2

s sin s sinsin s

2 2

tco t co x


t co t

2 2 2 2

3 3 220 0 0 0

4sin 4 s 4 42

sin sin sin 24

x co xI I I dx dx dx dx

x cosx x cosx x cosx cos x

2 tan 2 2 4 224

0x I

BI TP T GII

1. Tm nguyn hmF(x) ca hm sf(x)= sin cossin cos

x x

x x

, bit rng ln24

F

2. Tnh cc tch phn sau:A=

2

1

2 5- 7e x xdx

x

B=

22

-2

-1x dx C=2

0

2 ln 2x dx


33cos

0

sinxe xdx

B=4

1

lne xdx

xC*=

2 3

25 4

dx

x x D

*=2

1 1 -1

xdx

x

4. Tnh cc tch phn sau:I=

1

sin(ln )e xdx

xJ=

4

2

6

sin cot

dx

x x

K=10

1

lg xdx L=ln 5

ln 3 2 3x x

dx

e e

M=2

2 20

sin2

cos 4 sin

xdx

x x

N=

2

21 - 9

dx

xC=

2

2 20

sin2(1 cos )

xdx

x


1

20 4 -

dx

x B=

3

23 3

dx

x C=

42

0

16- dxx D=ln 2

0

1-1

x

x

edx

eE=

3

22

2

1dx

x

6.

Tnh cc tch phn sau:A=

2

1

lne xdx B

*=2

0

sin1 cosx x

dx

C*=

2

21

lndx

x

D*=1

cos(ln )e

x dx

E=2 4

31

3 2x xdx

x

1 2*

41

1

1

xF dx

x

7. Tnh:


53/108


53

A=4

2

0

cos xdx

B=2

3

0

cos xdx

C=1

0

xxe dx D=

4

1

xedx

x E=

2

1

lnx xdx

F=1

ln 1e xdx

x

G=

22

0

1 2x x dx H=4

0

1 2x xdx I=2

1 1

xdx

x J=

1

20 1

xdx

x

8. Tnh din tch hnh phng gii hn bi cc ng sau:a.x=1;x=e;y=0 vy= 1 ln x

x

b.y=2x;y=3x vx=0

c.y=sin2xcos3x, trc Ox vx=0,x=3

.

9. Tnh din tch hnh phng gii hn bi cc ng: y=0, y=x32x2+4x3 (C) v tip tuyn ving cong (C) ti im c honh bng 2.

10.Cho hnh phngD gii hn bi cc ngy=tanx,x=0,x=/3,y=0.a. Tnh din tch hnh phngD.b. Tnh th tch vt th trn xoay sinh ra bi hnh phngD quay quanh trc Ox.

11.Tnh th tch vt th trn xoay sinh ra bi hnh phng gii hn bi ng congy2=x3 vy=0,x=1 khi n quay quanh:

a)Trc Ox.b)Trc Oy.

TUYEN TAP TCH PHAN QUA CAC K THI AI HOC

A12. 32

1

1 ln 1xI dx

x

B12.

1 3

4 20 3 2

xI dx

x x

D12.

4

0

1 sin 2I x x dx

A11. 4

0

sin 1 cos

sin cos

x x xI dx

x x x

B11.

3

20

1 sin

cos

xI dx

x

D11.

4

0

4 1

2 1 2

xI dx

x

A10.1 2 2

0

21 2

x x

xx e x eI dx

e

B10. 21ln

2 ln

e

xI dxx x

D10.1

32 lne

I x xdxx

A09. 2

3 2

0

os 1 osI c x c xdx

B09.

3

20

3 ln

1

xI dx

x

D09.

3

1 1x

dxI

e

A08.46

0

tan

os2

xI dx

c x

B08. 4

0

sin4

sin 2 2 1 s inx cos

x dx

Ix

D08.2

31

ln xI dx

x

A07. Tnh din tch hnh phng gii hn bi cc ng 1y e x v 1 xy e x .

B07. Cho hnh phng H gii hn bi cc ng: ln , 0,x x y x e .Tnh th tch ca khi trn xoay to thnh khi H quay quanh trc Ox.

D07.2

3 2

0

lnI x xdx


54/108


54

A06.2

2 20

sin2

os 4sin

xdx

c x x

B06.

ln5

ln3 2 3x x

dx

e e D06.

12

0

2 xx e dx

A05.2

0

sin 2 s inx

1 3cos

xdx

x

B05.

2

0

sin 2 cos

1 cos

xdx

x

D05.

2sinx

0

cos cose x xdx

A04.2

1 1 1

xdx

x B04.

1

1 3ln .lne x xdx

x

D04.

32

2

ln x dx

A03.2 3

25 4

dx

x x B03.

24

0

1 2sin

1 sin 2dx

x

D03.

22

0

x x dx

A02. Tnh din tch hnh phng gii hn bi cc ng: 2 4 3 , 3x x y x

B02. Tnh din tch hnh phng gii hn bi cc ng:2 2

4 ,4 4 2

xy y

D02. Tnh din tch hnh phng gii hn bi ng cong3 1

1y

v hai trc ta .

MT S CAO NG V D B H

Bi 1. Tham kho 2005 dxx

xI

7

03 1

2; KQ:

14110

Bi 2. Tham kho 2005 3

0

2sin

xtgxdxI KQ: 3ln 28


sin

0

tan .cosxI x e x dx

KQ:12ln 2 e 1

Bi 4. Tham kho 2005 e

xdxxI1

2 ln KQ: 32 1e9 9

Bi 5. C Khi A, B 2005 dxxxI 1

0

23 3. KQ:6 3 8

5

Bi 6. C Xy Dng S 3 2005

3

1 313

3 dx

xx

xI KQ: 6 ln3 8

Bi 7. C GTVT 2005 dxxxI 1

0

25 1 KQ:8

105

Bi 8. C Kinh T K Thut I 2005 2

0

3 5sin

xdxeI x KQ:

323.e 5

34


55/108


55

Bi 9. C Ti Chnh K Ton IV 2005 dxxxI 53

0

3 .1 KQ:848105

Bi 10. C Truyn Hnh Khi A 2005

4

0

2

2sin1

sin21

dxxxI KQ:

1 ln 22

Bi 11. CSP Tp.HCM 2005

0

12 42xx

dxI KQ:3

18

Bi 12. C KT-KT Cn Th 2005 e

dxx

xI1

2

lnKQ:

21e

Bi 13. CSP Vnh Long 2005 dxx

xI

3

7

03 13

1KQ:

4615

Bi 14. C Bn Tre 2005

2

0 1sin

3cos

dxx

xI KQ: 2 3ln 2

Bi 15. CSP Sc Trng Khi A 2005

3

02

2

2

0 22

cos2sin

sin

2cos.cos2sin

sin

xx

xdxxJ

xxx

xdxI

KQ:

I ln2

3J

3 4

Bi 16. C Cng ng Vnh Long 2005 e

xdxxI1

ln KQ:2e 14

Bi 17. C Cng Nghip H Ni 2005 dxxxI sin4

0

2

KQ:2

42

Bi 18. CSP H Ni 2005 dxx

xxxI

2

02

23

4

942KQ: 6

8

Bi 19. C Ti Chnh 2005 1

031x

xdxI KQ:

18

Bi 20. CSP Vnh Phc 2005

e

xx

dxI1

2ln1KQ:

6

Bi 21. CSP H Ni 2004

2

020042004

2004

cossin

sin

dxxx

xI KQ:

4


56/108


56

Bi 22. CSP KonTum 2005

2

0

3

cos1

sin4

dxxx

I KQ: 2

Bi 23. Tham kho 20066

2

dxI

2x 1 4x 1

KQ:3 1ln2 12


0

I x 1 sin 2x dx

KQ: 14


1

I x 2 ln x dx KQ:5 ln 44


5

dxIx 2 x 1

KQ: 2 ln 2 1


e

1

3 2 ln xI dxx 1 2 ln x

KQ:

10 1123 3

Bi 28. C KTKT Cng Nghip II 2006 1

2

0

I x ln 1 x dx KQ:1ln 22

(i bin 2t 1 x , tng phn)

Bi 29. C C Kh Luyn Kim 2006 2

21

ln 1 xI dx

x

KQ:33ln 2 ln32

Bi 30. C Nng Lm 20061

2

0

I x x 1dx KQ:2 2 1

3

Bi 31. H Hi Phng 2006 1 20

xI dx1 x

KQ: 1 ln 22

Bi 32. C Y T 2006

2

4

sinx cosxI dx

1 sin 2x

KQ: ln 2

Bi 33. C Ti Chnh K Ton 2006 3

2

0

I x ln x 5 dx KQ: 1 14ln14 5ln5 92

Bi 34. C S Phm Hi Dng 2006

23

0

cos2xI dxsin x cosx 3

KQ: 132

Bi 35. H CH Hng Vng 2006 4

0

I x 1 cosx dx

KQ:2 1

8


57/108


57

Bi 36. C KTKT ng Du 20064

0

cos2xI dx1 2sin 2x

KQ:1 ln 34

Bi 37. C S Phm Qung Bnh 2006ln2 2x

x

0

eI dx

e 2

KQ:82 33

Bi 38. C S Phm Qung Ngi 200632

0

4sin xI dx1 cosx

KQ: 2

Bi 39. C S Phm Tr Vinh 20064

20

xI dxcos x

KQ:2ln

4 2

Bi 40. C Cng Ngh - Tp.HCM 20063

1

x 3I dx3 x 1 x 3

KQ: 6 ln3 8

Bi 41. C S Phm Tin Giang 20069

3

1

I x. 1 x dx

KQ:

468

7

Bi 42. C Bn Tre 2006e 3

1

x 1I ln x dxx

KQ:

32e 119 18

Bi 43.1

2 3

0

I x 2 x dx KQ: 2 3 3 2 29

Bi 44. 2

0

2cos12

xdxxI KQ:21

12 4 2

Bi 45. 1

0

32

1 dxxexIx

KQ:

2e 14 14

Bi 46. C KT-KT Cng Nghip I 20062

0

sin3xI dx2cos3x 1

KQ: Khng tn ti

Bi 47. C KT-KT Cng Nghip II 2006 1

2

0

I xln 1 x dx KQ:1ln 22

Bi 48. C Xy dng s 2 2006

2

1

x x 1I dxx 5

KQ:32

10ln33

Bi 49. C Xy dng s 3 2006 1

3

0

I x cos x sin x dx KQ:54

Bi 50. C GTVT III 2006


58/108


59/108


59

Bi 62. C Ti Chnh Hi Quan 2006 3

4

ln tgxI dx

sin2x

KQ: 21 ln 3

16

Bi 63. C K thut Cao Thng 2006

2 3

20I sin2x 1 sin x dx

KQ:154

Bi 64. CKT Tp.HCM Kha II 2006

e

0

lnxI dxx

KQ: 4 2 e

Bi 65. CCN Thc phm Tp.HCM 20061

20

1I dxx 2x 2

KQ:4

Bi 66. C in lc Tp.HCM 2006

73

30

x 2I dx

3x 1

KQ:4615

Bi 67. C Kinh t cng ngh Tp.HCM Khi A 2006:4

20

xI dxcos x

KQ:2ln

4 2

Bi 68. C Kinh t cng ngh Tp.HCM Khi D1 2006

2

1

I 4x 1 lnxdx KQ: 6 ln 2 2

Bi 69. CSP H Ni Khi D1 20063

6

dxIsinx.sin x

3

KQ:2 ln 23

.

Bi 70. Tham kho khi A 2007 I =4

0

2x 1dx

1 2x 1

KQ: 2 ln2

Bi 71. Tham kho khi B 2007: Tnh din tch hnh phng gii hn

bi cc ng

2

10

1

x xy v y

x. KQ:

1ln 2 1

4 2

Bi 72. Tham kho khi B 2007

Tnh din tch hnh phng gii hn bi cc ng 2 2 2 y x v y x . KQ:1

2 3

Bi 73. Tham kho khi D 2007 I = 1

2

0

x x 1dx

x 4

KQ:3

1 ln2 ln3

2

Bi 74. Tham kho khi D 2007 I =2

2

0

x cos x dx

KQ:2

24

Bi 75. CSPTW 2007Tnh din tch hnh phng gii hn bi cc ng c phng trnh

2y x 2 ; y x ; x 1; x 0 . KQ:7

6


60/108


60

Bi 76. C GTVT 2007 I =32

0

4 cos xdx

1 sin x

KQ: 2

Bi 77. CDL Cng ngh thng tin Tp.HCM 2007 I =7

3

0

x 2dx

x 1

KQ:

231

10

Bi 78. C Khi A 2007 I =20071

21

3

1 11 dx

x x

KQ:

2008 20083 2

2008

Bi 79. C C kh luyn kim 2007 I = e

2

1

x ln x dx KQ: 31 5e 2

27

Bi 80. CSP Vnh Phc 2007 I = 4

2

1

x s in x dx

KQ:3 2 1

384 32 4

Bi 81. C Khi B 2007: Tnh din tch hnh phng gii hn bi cc ng

y x , 2y x cos x , x 0 , x . KQ:2

Bi 82. C Khi D 2007 I =0

2

x 1 dx

KQ: 1

Bi 83. C Dt may thi trang Tp.HCM 2007

3

2 21

dx

x x 1KQ:

31

3 12

Bi 84. C Hng hi 2007 I =3

3 2

1

x x 1dx

KQ:

14 3

5

Bi 85. C Kinh t k thut Thi Bnh 2007

0

2x

1

x e x 1 dx

KQ:23 31e

4 60

Bi 86. D b 1 A_083

31

2

2 2

dx

x

D b 2 A_08.

2

0

sin2

3 4sin os2

xdx

c x

Bi 87. D b 1 B_08.2

0

1

4 1

xdx

x

D b 2 B_08.

1 3

20 4

xdx

x

Bi 88. D b D_08.1

2

20 4

x xe dxx

.

-----------------------------------------------------------------------------------------------------TIP THEO..


61/108


61

Cu III (1,0 im). Tnh tch phn:ln3 2

ln 2 1 2

x

x x

e dxI

e e

* t t = 2xe , Khi x = ln2 t = 0 x = ln3 t = 1 ex = t2 + 2 e2x dx = 2tdt

* I = 2

1 2

20

( 2)

1

t tdt

t t

= 2

1

20

2 1

( 1 )1

t

t dtt t

= 2

1

0 ( 1)t dt + 2

1 2

20

( 1)

1

d t t

t t

= 2 1( 2 ) 0t t + 2ln(t

2 + t + 1) 10 = 2ln3 - 1

Cu III(1 im). Tnh tch phn: I

4

02

211

1dx

x

x.

t dttdxx

dxdtxt )1(

21211

v

2

22 ttx

i cn

Ta c I = dttt

tdtt

tttdt

t

ttt

4

22

4

2

4

22

23

2

2 243

2

1243

2

1)1)(22(

2

1=

ttt

t 2ln43

22

1 2=

4

12ln2

Cu III(1 im) Tnh tch phn: I =2

30

sinxdx

(sinx + cosx)

t x =2

u dx = - du. i cn: x = 0 u =2 ; x =

2 u = 0

Vy: I =

2 2

3 30 0

sin( ) cosxdx2sinx+cosx

sin os2 2

u du

u c u

Vy : 2I =

2 2

2 20 0

s inx + cosx

(sinx + cosx)sinx + cosx

dxdx

=2

20

tan4

1222 os 0

4

xdx

c x

Cu III (1,0 im). ChoI=ln 2 3 2

3 20

2 1

1

x x

x x x

e edx

e e e. Tnh eI

I=ln2 3 2

3 20

2 1

1

x x

x x x

e edx

e e e=

ln 2 3 2 3 2

3 20

3 2 ( 1)

1

x x x x x x

x x x

e e e e e edx

e e e=

ln2 3 2

3 20

3 21

1

x x x

x x x

e e edx

e e e

X 0 4T 2 4


62/108


62

= ln(e3x + e2x ex + 1)ln 2 ln 2

0 0 x = ln11 ln4 =

14ln

4; Vy eI =

11

4.

Cu III(1,0 im): Tnh tch phn sau.3

2 4

4

sin .cos

dxI

x x

x =

= 3

Khi 3 438)321()21()1(

3

1

33

1

22

3

12

22

tttdtttt dttI

Cu III: (1 im). Tnh tch phn :2

3

2

1

ln( 1)xI dx

x

.

t

22

32

2ln( 1)

11

2

duu xx

dxdv

vx

Do I =22

2 21

2ln( 1)

12 ( 1)

x dx

x x x

2

21

ln 2 ln 5 1

2 8 1

x

dxx x

2 2 2

21 1

ln 2 ln 5 1 ( 1)

2 8 2 1

dx d x

x x

2 2ln 2 ln 5 1ln | | ln | 1 |

12 8 2x x

=

52ln 2 ln 5

8

Cu III (1 im). Tm nguyn hm xdx

I53 cos.sin

dx

x

dxI

23233 cos.2sin8

cos.cos.sint tanx = t

2 3

2 2 332

2 ( 1); sin 2 ; 8

2cos 1 ( )1

dx t dt t

dt x I dt tx t tt

Cx

xxxdttt

tt

dtt

ttt

2

2433

3

246

tan2

1tanln3tan

2

3tan

4

1)

33(

133

3

4

42 cos.sin

xx

dxI

3

4

22 cos.2sin.4

xx

dx

t : t = tanx =

i cn: x =

= 1


63/108


64/108


64

Cu III (1.0 im) . Tnh tch phn:23

1

ln 2 lne x xI dx

x

23 12 2 23 3

1 1 1

4234 433

1

ln 2 ln 1ln 2 ln ln 2 ln 2 ln

2

3 2 ln1 3. 3 2

2 4 8

e e e

e

x xdx x xd x x d x

x

x

Cu III (1.0 im) . Tnh tch phn :/2 2

/6

sin

sin3B dx

x

/ 2 /2 /2 /2 /22 2 2

3 2 2 2/6 /6 /6 /6 /6

sin sin sin sin sin

sin 3 3sin 4sin sin (3 4sin ) 3 4sin 4cos 1

x x x x xdx dx dx dx dx

x x x x x x x

t t = cosx => - dt = sinxdx . Ta c :

0 3 /2 3 /2

2 20 03/ 2

1 1 1... ln(2 3)

4 1 4 1/ 4 4 ( 1 / 2)( 1/ 2) 4

dt dt dt B

t t t t

.

Cu III (1,0 im). Tnh tch phn3

0

3

3. 1 3

xdx

x x

.

t u = 21 1 2x u x udu dx ; i cn:0 1

3 2

x u

x u

Ta c:3 2 2 23

20 1 1 1

3 2 8 1(2 6) 6

3 2 13 1 3

x u udx du u du du

u u ux x

22

1

26 6ln 1

1u u u

33 6 ln

2

BI TP TIP THEOBi 1. Tnh cc tch phn sau

a.1

2

0

x x 1dx b.2

2x 2

1

3e dx

x

c.

2

21

xdx

x 2

d.1 3 2

22

(x 4)dx

x

e.

3

22

dx

x 4x 4 f.

2e

1

2 x 5dx

x

g. 2

0

2sin xdx

h.4

20

1 sin xdx

cos x

i.

5

2

dx

x 2 x 2

Bi 2. Tnh cc tch phn sau


65/108


65

a.2

21

dx

(x 4x 4) b.

2 2011

20111

1 xdx

x(1 x )

c.

1 2

2

x 2xdx

x 3

d.1

20

dx

x 5x 6 e.

0

21

dx

x 2x 2

f.

3 2

2

x 5x 4dx

x 1

g.2

31

dxx x

h.3 3

20

x dxx 2x 1

i.2

21

dxx (1 x)

j.3 2

32

x 3x 2dx

x 3x 2

k.

1 2

30

x dx

(x 2)l.

1

2 20

dx

(x 2) (x 1)

m.1

40

xdx

x 1n.

2

41

dx

x(x 1)


a./4

20

tanxdx

cos x

b./3

2

/4

3tan xdx

c./2

0

dx

1 sin x

d./2

0

1 cos xdx

1 cos x

e.

3

6

(tanx cot x)dx

f.2

2

sin x cos xdx

sin x cos x


a.ln 2

x0

dx

e 5b.

ln3

x0

1dx

e 1 c.

ln 22x x

0

e e 1dx

d.ln 2 x x

x x0

e edx

e e

e.1 2x

x0

edx

e 1

f.e

21

ln xdx

x(ln x 1)


a.2

x

0

e sin xdx

b.1

2x

0

xe dx c.1

0

xln(1 x)dx

d.3

0

ln(x 1)dx

x 1

e.

e

21

ln xdx

xf.

3

2

6

ln(sin x)dx

cos x


a.

119

0 x(1 x) dx

b.

1 3

2 30

x

dx(1 x ) c.

1

0

xdx

2x 1

d.1

3 2

0

x 1 x dx e.2 5

25

dx

x x 4 f.

3 5 3

20

x 2xdx

1 x

g.ln3 x

x 30

edx

(1 e ) h.

2

2 20

sin2xdx

cos x 4sin x

i.

32

20

cosx sin xdx

1 sin x


66/108


66


a.

1

2

20

dx

1 x b.

1 3

20

xdx

4 x c.

22 2

1

x 4 x dx

d.

1

4 20

2dx

x 4x 3 e.1

4 20

xdx

x x 1 f.0

21

dx

x 2x 2

g.2 2

31

x 1dx

x

h.

1

320

dx

1 x i.

222

20

xdx

1 x

j.2

2

0

x 2x x dx k.2 3 3

41

x xdx

x

l.

3

2

33

xdx

x 1

PHNG PHP TCH PHN TNG PHNTnh cc tch phn sau

a.4

2

0

x tan xdx

b.2

cosx

0

e sin 2xdx

c.e

3

1

ln xdx

d.1

3 2

0

x ln(x 1)dx e.3

2

2

ln(x x)dx f.6

2

0

xsin xcos xdx

g.2

3

cos x ln(1 cos x)dx

h.2

2

1

1x ln(1 )dx

x g.

1

2

0

1 xln dx

1 x

TCH PHN C CHA DU TR TUYT IBi 1. Tnh cc tch phn sau

a.2

2

0

x 2x 3 dx b. 4

2

x 2 x 2 dx

c.3

x

0

2 4dx


a.4

0

2 2 cos 2xdx

b.2

0

1 sin xdx

c.3

2 2

6

tan x cot x 2dx

d.3

3

6

cos x cos x cos xdx

e.

0

cos x sin xdx

f.

e

1

e

lnx dx

TCH PHN CC HM S LNG GICBi 1. Tnh cc tch phn sau

a.2

3

0

sin xdx

b.2

2 4

0

sin x cos xdx

c.2

4 5

0

sin x cos xdx


67/108


67

d.2

0

sin2xcosxdx

1 cos x

e.

43

0

tan xdx

f.3

3

4

dx

sin x cos x

g.

32

20

sin xdx

1 cos x

h.32

0

cos xdx

1 cos x

f.3

4

6

dx

sin x cos x


a.2

3 5

0

1 cos x sin x cos xdx

b.2

6

1 sin 2x cos 2xdx

sin x cos x

c.

3

2

4

tan xdx

cos x 1 cos x

d.2

4 4

0

cos 2x(sin x cos x)dx

e.2

2 3

0

(1 sin x) sin 2xdx

f.34

0

sin xdx

cosx

g.4

20

dx

1 8cos x

h.2

6

dx

sinx

i.

2

0

dx

2 cos x


a.3

6

dx

sin xsin(x )6

b.

2

0

sinxdx

2 sin x

c.

2

0

dx

1 sin x cos x

d.2

0

sin x cos x 1dx

1 sin x cos x

e.

4

0

xdx

1 cos 2x

(tng phn) f.

2

0

sinxdx

sin x cos x


a.2

3

0

sin xdx

b.2

2x 1

0

sin 2xe dx

c.2

1

cos(ln x)dx

d.3

2

6

ln(sin x)dx

cos x

(tng phn) e.

2sin x 3

0

e sin xcos xdx

f.4

40

dx

cos x

BI TP TNG HP V TCH PHNBi 1. Tnh cc tch phn sau

a.3 7

4 82

xdx

1 2x x b.

22

1

x 1dx

x 2

c.

2 3 2

20

x 2x 4x 9dx

x 4


a.1

20

dx

2x 5x 2 b.

1

30

xdx

(x 1)c.

9 2

32

x dx

x 1


68/108


68

d.2

1

xdx

x x 1 e.

4

1

2dx

x 5 4 f.

15 2

0

x 1 x dx


a.2

31

1dx

x 1 x b.

2

1

xdx

2 x 2 x c.

3

1

x 3dx

3 x 1 x 3

d.1 2

230

x xdx

(x 1)

e.

33 2

0

x x 1dx f.2

2

0

4 x dx

g.6

34

x 4dx

(x 2)

h.

3

0

1dx

4 xi.

2 2

41

4 2xdx

4 x

j.1

5 3 6

0

x (1 x ) dx k.2 5

51

1 4xdx

x(1 x )

l.

3

6 21

1dx

x (1 x )


a.2

2

0

ln( 1 x x)dx b.1 2 x

20

x e dx(x 2) c.

2

1

3 x

0

x e dx

d.1

2x x0

1dx

e ee.

ln 2x

0

e 1dx f.1

20

ln(1 x)dx

x 1

g.

2e

2e

1 1dx

ln x ln x

h.4

x

1

e dx


a.3

2 2

6

(tan x cot x)dx

b.

24

0

tan x 1dx

tan x 1

c.

4

30

1dx

cos x

d.2

40

sin2xdx

1 cos x

e.

3

2 2

3

1dx

sin x 9cos x

f.33

6

4sin xdx

1 cos x

g.2

20

sin2xdx

(2 sin x)

h.

2

0

sin 2x sin xdx

1 3cos x

i.

2

4

0

cos xdx

j.

2

0

sin3x

dxcos x 1

k.

2

20

cosx

dxcos x 1

l.

4 44

0

sin x cos x

dxsin x cos x 1

m.36

0

sin x sin xdx

cos2x

n.

2

3

1dx

sin x 1 cos x



69/108


69

a.1

2 3

0

(1 x ) dx b.1

33

1dx

x 4 (x 4)

c.

6

22 3

1dx

x x 9

d.1

2

0

1 6x 3x dx

Phng php tnh tch phn CC HM S LNG GIC V V T

TCH PHN CHA CC HM S LNG GICI. KIN THC1. Thuc cc nguyn hm :

a/ 1

sin ax+b os ax+bdx ca

b/

sin ax+b

ln os ax+bos ax+b

dx cc

c /

1os ax+b sin ax+bc dx

a

d/

os ax+bln sin ax+b

sin ax+b

cdx

2. i vi : ( )I f x dx

a/ Nu f(x)= nsin ; osmR x c x th ta ch :- Nu m l , n chn : t cosx=t ( Gi tt l l sin )- Nu n l , m chn : t sinx=t ( Gi tt l l cos )- Nu m,n u l th : t cosx=t hoc sinx =t u c ( gi tt l sin hoc l cos )- Nu m,n chn : t tanx=t ( gi tt l chn sinx , cosx )b/ Phi thuc cc cng thc lng gic v cc cng thc bin i lng gic , cc hng ng thclng gic , cng thc h bc , nhn i , nhn ba , tnh theo tang gc chia i ....

3. Ni chung tnh c mt tch phn cha cc hm s lng gic , hc sinh i hi phi c mts yu t sau :- Bin i lng gic thun thc- C k nng kho lo nhn dng c cch bin i a v dng bit trong nguyn hm .

II. MT S V D MINH HAV d 1. Tnh cc tch phn sau :

a. (H, C Khi A 2005)

2

0 cos31

sin2sin

dxxxxI

b..H, C Khi B 2005 . dxx

xxI 2

0 cos1cos2sin

KQ: 2 ln 2 1

Gii

a.

2 2

0 0

2cos 1 s inxsin 2 sin1

1 3cos 1 3cos

xx xI dx dx

x x


70/108


70

t :

2t 1 2osx= ;s inxdx=-

3 31 3cos

0 2; 12

c tdt

t x

t x t

Khi :

2

1 2 23

2 1

12 1 23 2 2 1 2 1 342

13 9 9 3 27

tt

I tdt dt t tt

b. 2 22 2 2

0 0 0

sin 2 cos 2sin cos os2 s inxdx 1

1 cos 1 cos osx+1

x x x x c xI dx dx

x x c

t :

2

dt=-sinxdx, x=0 t=2;x= 12

1 osx1 1

( ) 2

t

t ct

f x dx dt t dt

t t

Do :12

2

0 2

21 12 ( ) 2 2 2 2 ln 2 ln 2 1

12I f x dx t dt t t t

t

V d 2. Tnh cc tch phn sau

a. H- C Khi A 2006 .2

2 20

sin2xI dxcos x 4sin x

KQ:23

b. C Bn Tre 2005 . 2

0 1sin

3cos

dxx

x

I KQ: 2 3ln 2 Gii

a.2

2 20

sin2xI dxcos x 4sin x

. t :2 2 2 2 2os 4sin os 4sint c x x t c x x

Do :

22 2sin cos 8sin cos 3sin 2 sin 2

3

0 1; 22

tdt x x x x dx xdx xdx tdt

x t x t

Vy :2 22

0 1 1

22 2 2 2( )

13 3 3 3

tdtI f x dx dt t

t

b.

2

0 1sin

3cos

dxx

xI .

Ta c : 3 2 2 2os3x=4cos 3cos 4cos 3 osx= 4-4sin 3 osx= 1-4sin osxc x x x c x c x c


71/108


71

Cho nn :

21 4sinos3x

( ) osxdx 11+sinx 1 s inx

xcf x dx dx c

t :

2

dt=cosxdx,x=0 t=1;x= 22

1 s inx1 4 1 3( ) 8 4

t

tt

f x dx dt t dtt t

Vy : 22

2

0 1

23( ) 8 4 8 2 3ln 2 3ln 2

1I f x dx t dt t t t

t


a. CSP Sc Trng Khi A 2005 .2

2 20

sin

sin 2cos .cos2

xdxI

x x

b. C Y T 2006 .

2

4

sinx cosxI dx

1 sin 2x

KQ: ln 2

Giia.

2 2 2

22 20 0 0

sin sin s inxln 1 osx ln 22

sin cos . 1 osx 1+cosxsin 2cos .cos 02

xdx xdxI dx c

x x x cx x

b.

2 2 2

2

4 4 4

sinx cosx sinx cosx sinx cosxI dx dx dx 1

sinx+cosx1 sin 2x sinx+cosx

V : s inx+cosx= 2 sin ; 3 sin 04 4 2 2 4 4 4

x x x x

Do : sinx+cosx sinx+cosx

Mt khc : s inx+cosx osx-sinxd c dx

Cho nn : 2

4

sinx+cosx 12ln s inx+cosx ln1 ln 2 ln 2sinx+cosx 2

4

dI


a. C S Phm Hi Dng 2006 .

2

30

cos2xI dxsin x cos x 3

KQ:1

32


72/108


72

b. C KTKT ng Du 2006 .4

0

cos2xI dx1 2sin 2x

KQ:1 ln 34

Gii

a.

2

30

cos2xI dxsin x cos x 3

. V : 2 2cos 2 os sin osx+sinx osx-sinxx c x x c c

Cho nn :

3 3

osx-sinxos2x( ) osx+sinx

sinx-cosx+3 sinx-cosx+3

ccf x dx dx c dx

t :

3 2 3

dt= cosx+sinx ; 0 2, 42

sinx-cosx+33 1 1

( ) 3

dx x t x t

tt

f x dx dt dtt t t

Vy : 422 3 2

0 2

41 1 1 3 1 1( ) 324 32

I f x dx dtt t t t

b.4

0

cos2xI dx1 2sin 2x

. t :

14cos 2 os2xdx=

41 2sin 2

0 1; 34

dt xdx c dt

t x

x t x t

Vy :

34

0 1

3cos2x 1 dt 1 1I dx ln t ln31 2sin 2x 4 t 4 1 4

V d 5. Tnh cc tch phn sau :

a. C S Phm Qung Ngi 2006 .32

0

4 sin xI dx1 cosx

KQ: 2

b. C Bn Tre 2006 .36

0

sin3x sin 3xI dx1 cos3x

Gii

a.

232 2 22

0 0 0

1 cos x4sin x 1

I dx 4 s inxdx=4 1 cosx s inxdx=4. 1 cosx 221 cosx 1 cosx 2 0

b.36

0

sin3x sin 3xI dx1 cos3x

.

Ta c : 3 2 2sin 3 sin 3 sin 3 1 sin 3 sin 3 . os 3x x x x c x .


73/108


73

t :

1dt=-3sin3xdx sin3xdx=-

31 os3x

0 2; 16

dt

t c

x t x t

Vy : 21 26

2

0 2 1

211 1 1 1 1 1 1( ) 2 2 ln ln 213 3 3 2 6 3

tf x dx dt t dt t t t

t t


a. I =3 32

3

sin x sin xcotgxdx

s inx

b. I =

2

2

s i n ( x )4 d x

s i n ( x )4

c. I =2

4

0

sin x dx

d. I = dxxxnsix )cos(2cos

442

0

Gii

a. I =

33 3 22 2

3 3

1s inx 1

sin x sin x sin xcot gx dx cot xdx

sin x s inx

2 23 23

2

3 3

11 cot xdx cot x cot xdx

sin x

b. I =2 2

2 2

s i n ( x ) c o s x - s i n x4 d x d xc o s x + s i n xs i n ( x )

4

2

2

d cosx+sinx 2ln cosx+sinx 0cosx+sinx

2

c. I =

22 2 24

0 0 0

1 cos2x 1 1 cos4xsin x dx dx 1 2cos 2x dx

2 4 2


74/108


75/108


75

c. I = 3 3 3

22 2

6 6 6

tg x cot g x 2dx t anx-cotx dx t anx-cotx dx

V :2 2

sinx osx sin os os2xtanx-cotx= 2 2cot 2cosx sinx s inxcosx sin2x

c x c x c

Cho nn :

t anx-cotx0;x ;4 3

x x x

Vy : 3 34 4

6 4 6 4

os2x os2x 1t anx-cotx t anx-cotx

sin2x sin2x 2

c cI dx dx dx dx

14 3ln sin 2 ln sin 2 ln 22

6 4

x x

d. I =2

3 3

0

( cos x sin x )dx

(1)

t : , 0 ; 02 2 2x t dx dt x t x t

Do :

0 2 2

3 3 3 33 3

0 02

os sin sin ost sin osx 22 2

I c t t dt t c dt x c dx

Ly (1) +(2) v vi v : 2 0 0I I V d 8 . Tnh cc tch phn sau

a.3

4

4

tan dx

(Y-HN-2000) b.

4

0

os2x

sinx+cosx+2

cdx

(NT-2000) c.62

4

4

os

sin

c xdx

x

(NNI-2001)

d.24

60

sin

osdx

c x

( GTVT-2000) e.2

20

sin2

4 os

xdx

c x

f.

24

0

1 2sin

1 sin 2dx

x

(KB-03)

Gii


76/108


76

a.3

4

4

tan dx

. Ta c :

224

44 4 4 2

1 ossin 1 1( ) tan 2 1

os os os os

c xxf x x

c x c x c x c x

Do : 3 3 3

24 2 2

4 4 4

1 1 3( ) 2 1 1 tan 2 tanos os os

4

dxI f x dx dx x x xc x c x c x

31 4 23t anx+ tan 2 3 2 2 3 2 3 23 12 3 12 3 12

4

x

* Ch : Ta cn c cch phn tch khc :

4 2 2 2 2 2

2 2 2

( ) tan tan tan 1 1 tan 1 tan tan

tan 1 tan tan 1 1

f x x x x x x x

x x x

Vy : 3 3 3 3

2 2 2 22 2

4 4 4 4

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Bai Tap Tich Phan Day Du LTDH