(0; )a b 1 sin( )

sin( )sin( ) sin( ) sin( )sin( )

dx a bI dx

x a x b a b x a x b

1 sin[( ) ( )]

sin( ) sin( )sin( )

x a x bdx

a b x a x b

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

sin( ) sin( )sin( )

x a x b x a x bdx

a b x a x b

1 cos( ) cos( ) 1 sin( )[ ] ln

sin( ) sin( ) sin( ) sin( ) sin( )

x b dx x a x bdx c

a b x b x a a b x b

NẾU:

sin( )sin( )

dxI

x a x b

Dạng 1

NẾU 0; ( ) ( )a b x a x b

( ) ( ) sin( ) sin( )x a x b x a x b

2cot( )

sin ( )

dxI x a c

x a

os( ) os( )

dxI

c x a c x b

1 sin[( ) ( )]0;

sin( ) os( ) os( )

x a x ba b I dx

a b c x a c x b

1 sin ( ) sin( ) 1 os( )[ ln

sin( ) os( ) os( ) sin( ) os( )

x a x b c x bdx dx c

a b c x a c x b a b c x a

2

0; ( ) ( ) ( ) ( )

os( ) os( ) ( )os ( )

a b x a x b x a x b

dxc x a c x b I tg x b c

c x b

Dạng 2

sin( ) os( )

dxI

x a c x b

1 os[( ) ( )]

2 os( ) sin( ) os( )

c x a x b dxa b I

c a b x a c x b

1 os( ) sin( )

[ ]os( ) sin( ) os( )

c x a x bdx dx

c a b x a c x b

1

[ln sin( ) ln os( )os( )

x a c x b cc a b

1 sin( )ln

os( ) os( )

x ac

c a b c x b

2a b

Đưa I về nguyên hàm cơ bản

Dạng 3

, ( 1)s inx

dxI m

m

2 ( )

2s inx sin 2sin( ) os( )

2 2 2 2

xddx

x xc

sin( )1 2 2lnos os( )

2 2

x

cxc c

Dạng 4

( )I tgxtg x dx s inx sin( ) sin x sin( ) cos cos( ) cos cos( )

cos os( ) cos cos( )

x x x x x xdx dx

x c x x x

os os sin[ ( )]

cos cos( ) sin( ) cos cos( )

c c x xdx dx dx dx

x x x x

cos( )

cot lncos

xx c

x

VÍ DỤ ( )4

I tgxtg x dx

sin x sin( ) sin x sin( ) cos cos( ) cos cos( )4 4 4 4

cos cos( ) cos cos( )4 4

x x x x x xdx dx

x x x x

Dạng 5

os sin[ ( )]24 42cos cos( ) sin( )cos cos( )

4 4 4

c x xdx dx dx

x x x x

cos( )4ln

cos

xx c

x

sin( ) os( )( ) cot( )

os( )sin( )

x c xI tg x x dx dx

c x x

sin( ) os( ) os( )sin( ) os( )sin( )

os( )sin( )

x c x c x x c x xdx

c x x

sin( )

sin( )os( )sin( ) sin( ) os( )

dxdx dx x

c x x x c x

Dạng 3

Dạng 6

Dạng7 otg(x+ )cotg(x+ )dxI c os( ) os( ) os[( ) ( )] sin( )sin( )

sin( )sin( ) sin( )sin( )

c x c x c x x x xdx dx

x x x x

os( )sin( )sin( )

dxc dx

x x

Dạng 1

Dạng 8a sin cos

dxI

x b x

2 2

2 2 2 2

1 x

s inx cos

da ba b x

a b a b

2 2

2 2

2 2

os1

,sin( )

s in

ac

dx a bbxa b

a b

2 2

1ln

2

xtg

a b

c

Dạng 9 a sin cos

sin cos

x b xI dx

c x d x

'a sin cos ( sin cos ) ( sin cos )

sin cos

x b x A c x d x B c x d x

c x d x

'( sin cos )ln sin cos

sin cos

c x d xI A dx B dx Ax c x d x c

c x d x

Viết

Dạng102

a sin cos

( sin cos )

x b xI dx

c x d x

'a sin cos ( sin cos ) ( sin cos )x b x A c x d x B c x d x

'

2

( sin cos )

sin cos ( sin cos )

dx c x d xI A B dx

c x d x c x d x

Viết

2 2ln

2

A xtg

a b

sin cos

BC

c x d x

Dạng11a sin cos

dxI

x b x c

2 2

2 2

2 2 2 2

1

s inx cos 1

dxc a b I

a ba b xa b a b

2 2 2 2

1 1

sin xsin cos os 1 os( ) 1

dx dx

xc c xa b a b

TH1

2 2

21 1 12 ( )2 22cos ( ) cos ( )

2 2

xddx x

tg Cx xc c c

TH2 2 2c a b

2 2

2 2 2 2

1

s inx cos 1

dxI

a ba b xa b a b

1 1

sin s inx cos cos 1 os( ) 1

dx dx

c x c c x

2 2

2 ( )1 1 12 cot( )21 2sin ( ) 1 2sin ( )

2 2

xddx x

Cx xc c c

TH3 2 2 2c a b

2

xt tg

2

2 2

2 1s inx ,cos

1 1

t tt

t t

2 2

2

1 1 1(1 ) (1 )

2 2 2 2os2

dx xdt tg dx t dx

xc

Dạng12 a sin cos

sin cos

x b x cI dx

m x n x p

ln sin cossin cos

dxAx m x n x p C

m x n x p

a sin cos

sin cos

x b x c

m x n x p

'( sin cos ) ( sin cos )

sin cos

A m x n x p B m x n x p C

m x n x p

'( sin cos )

sin cos sin cos

m x n x p dx dxI A dx B C

m x n x p m x n x p

Dạng132 2a sin sin x cos cos

sin cos

x b x c xI dx

d x e x

2 2a sin sin x cos cos ( s inx cos )( sin cos )x b x c x A B x d x e x

2 2(sin os )C x c x

( s inx cos )sin cos

dxI A B x dx C

d x e x

Dạng 8

Dạng14

2 2 2( ) os

dx dtgxI

atg x btgx c c x atg x btgx c

2 2a sin sin cos cos

dxI

x b x x c x

2

dt

at bt c

Dạng15 2 2 2 2

sin x cos

( sin os )

xdxI

a x b c x

2 2 2 2 2 2( sin os ) (2 sin x cos 2 cos sin )d a x b c x a x b x x dx 2 2 2 2

2 22 2

( sin os )2( )sin x cos sin x cos

2( )

d a x b c xa b xdx xdx

a b

Ta có

2 2 2 2 12 2

2 2 2 22 2

11: 1 ( sin os )

2( )( 1)

12 : 1 ln( sin os )

2( )

TH I a x b c x Ca b

TH I a x b c x Ca b

2 2 2 2

2 2 2 2 2 2

1 ( sin os )

2( ) ( sin os )

d a x b c xI

a b a x b c x

DẤU HIỆU ĐỔI BIẾN SỐ

(s inx,cos )f x dx(s inx, cos ) (sin ,cos ) s inxf x f x x t

( s inx,cos ) (s inx,cos ) cosf x f x x t

( s inx,-cos ) (s inx,cos )f x f x tgx t Mọi trường hợp trên đều đưa về tích phân hàm hữu tỉ bằng cách đặt

2

xt tg

CÁC BÀI TOÁNnI tg xdx

22

( ) (1 )os

dxtgx t d tgx dt dt t dx dt

c x

21

dtdx

t

21n dt

I tt

Ví dụ 6I tg xdx4

2

1( 1)

ostg x dx

c x 4 4( )tg xd tgx tg xdx

52

2

1( 1)

5 os

tg xtg x dx

c x

52 2( )

5

tg xtg xd tgx tg xdx

5 3

2

1( 1)

5 3 os

tg x tg xdx

c x

5 3

5 3

tg x tg xtgx x C

C2: 6I tg xdx2

22

64 2

2 2

5 3

os

(1 )1

1( 1 )

1 1

ar5 3

dxtgx t dt

c xdt

tg x dx dt dxt

tI dt t t dt

t t

t tt ctgt C

Ví dụ2os n

dxI

c x

2 2 2

1

os osn

dx

c c x 2 2

1( )

os nd tgx

c 1

2

1( ) ( )

os

n

d tgxc x

2 1(1 ) ( )ntg x d tgx

Ví dụ:6os

dxI

c x

4 2

1

os os

dx

c x c x 2

2

1( ) ( )

osd tgx

c x

2 2( 1) ( )tg x d tgx 4 2( 2 1) ( )tg x tg x d tgx

5 32

3tg x tg x tgx C

2 1os n

dxI

c x

2 2 2 1 2 1

cos sin sin

os ( os ) (1 sin )n n n

xdx d x d x

c x c x x

1 1

sin

(1 s inx) (1 s inx)n n

d x

1

1 1 1

1 [(1 s inx) (1 s inx)]

2 (1 s inx) (1 s inx)

n

n n ndx

VÍ DỤ

3cos

dxI

x

4

cos

cos

xdx

x 2 2

sin

(1 sin )

d x

x

2 2

sin

(1 sin ) (1 sin )

d x

x x

2

2 2

1 [(1 sin ) (1 sin )] (sin )

4 (1 sin ) (1 sin )

x x d x

x x

2 2

2 2

1 (1 sin ) 2(1 sin )(1 sin ) (1 sin )sin

4 (1 sin ) (1 sin )

x x x xd x

x x

VÍ DỤ

2 2

1 sin 1 sin 1 sin

4 (1 sin ) 2 (1 sin )(1 sin ) 4 (1 sin )

d x d x d x

x x x x

2

1 1 1 (1 sin ) (1 sin )sin

4 (1 sin ) 4 (1 sin )(1 sin )

1 (1 sin )

4 (1 sin )

x xd x

x x x

d x

x

1 1 1 1

ln 1 sin ln 1 sin4 (1 sin ) 4 4

1

4(1 sin )

x xx

Cx

C2: 3cos

dxI

x

2

2 2

1 sin

cos cos

cos cos

xu du dx

x xdx dx

dv v tgxx x

2 2

3 3

sin 1 cos

cos cos cos cos cos cos

tgx x tgx x tgx dxI dx dx I

x x x x x x

sin

cos (1 sin )(1 sin )

tgx d xI I

x x x

1 1 sin 1 1 sin2 ln ln

cos 2 1 sin 2cos 4 1 sin

tgx x tgx xI I C

x x x x

6 5: ; ;

cos os os

dx dx dxBT

x c x c x

2os nI c xdx2 1 os2

( os ) ( )2

nn c x

c x dx dx

VÍ DỤ

4osI c xdx 21 os2( )

2

c xdx

21 2cos 2 os 2

4

x c xdx

os2 1 1 os4

4 2 4 2

dx c x c xdx dx

3 1 1 4os2 (2 ) os4

8 4 8 4

d xdx c xd x c x

3 1 1sin 2 sin 4

8 4 32x x x C

2 1 2 2os ( os )cos (1 sin ) sinn n nI c xdx c x xdx x d x 5 2 2 2 2

2 4 3 5

: os ( os ) cos (1 sin ) sin

2 1(1 2sin sin ) sin s inx sin sin

3 5

VD c xdx c x xdx x d x

x x d x x x C

3 53

: os , sin 2 ,sin

dxBT c xdx xdx

x

2 2 12 1 2

, , sin , sinsin sin

n nn n

dx dxI dx dx

x x

46

sin 2 ,sin

dxxdx

x

3:sin x cos( )

4

dxD I

x

os( ) os[ ( )]14 41:2os( )sin x cos( ) sin x cos( )

4 4 42

c dx c x xC I dx

c x x

cos cos( ) sin x sin( )4 42

sin x cos( )4

x x xdx

x

sin( )cos 42 [ ] 2[ ln s inx ln os( )s inx 4os( )

4

xxdx c x C

c x

s inx2 ln

os( )4

Cc x

2

2

2 :1 sin x cos sinsin x cos( ) s inx(cos s inx)

4 2cot

2 2 2 ln cot 1sin (cot 1) cot 1

dx dx dxC I

x xx x

dx d xx C

x x x

Ví dụ:2sin 1

dxI

x

12(s inx )

2

dx

2(s inx sin )6

dx

1

4 sin( ) os( )2 12 2 12

dxx x

c

2 ( )1 24 sin( ) os( )

2 12 2 12

xd

x xc

sin( )1 1 2 12ln2 os os( )

6 2 12

x

Cx

c c

2sin 2 3

dxI

x

2cos 1

dxI

x

( ) ( ) (3 ) 3 (3 )3 3 4 4

I tgxtg x tg x tg x dx tg xtg x dx

sin 3 sin(3 ) os3 cos(3 ) os3 cos(3 )

4 4 4

os3 cos(3 )4

x x c x x c x xdx

c x x

os1 4 33 sin( ) os3 cos(3 )

4 4

cd x x

c x x

os( )4

os3 cos(3 )4

cdx dx

c x x

os1 4 33 sin( ) os3 cos(3 )

4 4

cd x x

c x x

cos(3 )1 4ln3 os3

xx C

c x

3 s inx cos

dxI

x

2

11:

( 3) 1 sin x cos cos sin6 6

dxC I

x

1

2 sin( )6

dx

x

1

2 2sin( ) os( )2 12 2 12

dxx x

c

2

1

4 ( ) os ( )2 12 2 12

dxx x

tg c

2 ( )1 2 124 ( )

2 12

xdtg

xtg

VÍ DỤ

1ln ( )

2 2 12

xtg C

Ví dụ 4sin 3cos

s inx 2cos

x xI dx

x

'4sin 3cos (s inx 2cos ) (s inx 2cos )x x a x b x

2 32; 1

2 4

a ba b

a b

's inx 2cos )2

s inx 2cos

x dxI dx

x

2 ln s inx 2cosx x C

Ví dụ 8cos

2 3 sin 2 cos 2

xI dx

x x

2

8cos

2 2 3 sin x cos 1 2sin

xdx

x x

2

8cos

( 3 s inx cos )

xdx

x

'8cos ( 3 s inx cos ) ( 3 s inx cos )x a x b x

( 3 )s inx ( 3)cosa b a b x 23 0

2 33 8

aa b

ba b

'

2

( 3 s inx cos )2 2 3

3 s inx cos ( 3 s inx cos )

dx x dxI

x x

J K

DẠNG10

23 s inx cos

dxJ

x

22(sin x cos cos sin )

3 3

dx

x

sin( )3

dx

x

2sin( ) os( )2 6 2 6

dxx x

c

2 ( )1 2 6 ln ( )2 2 6( )

2 6

xdtg x

tgx

tg

12 3

3 s inx cosK

x

1ln ( ) 2 3

2 6 3 s inx cos

xI tg C

x

VÍ DỤ

2 2

2

1 1 1(1 ) (1 )

2 2 2 2 2os2

x dx xt tg dt tg dx t dx

xc

2

2

2 2

21

2 12 1

1 1

dttIt tt t

2

2

2

dt

t t

2( 2)

dt

t t

1 ( 2)2

2 ( 2)

t tdt

t t

ln2 2

dt dt tC

t t t

2 2 2(2 ( 1) 1 ) 2s inx cos 1

dxJ

x

VÍ DỤ: 5s inx

2sin cos 1I dx

x x

'5s inx (2sin cos 1) (2sin cos 1)a x x b x x c

2 5 2

(2 )s inx (2 )cos 2 0 1

0 2

a b a

a b b a x a c b a b

a c c

'2(2sin cos 1) (2sin cos 1) 2

2sin cos 1

x x x xI dx

x x

'(2sin cos 1)

2 22sin cos 1 2sin cos 1

x x dx dxdx

x x x x

J

Ví dụ24sin 1

3 s inx cos

xI dx

x

2 2 24sin 1 (a sin cos )( 3 s inx cos ) (sin os )x x b x x c x c x

2 2( 3 )sin ( 3)sin x cos ( ) osa c x a b x b c c x

3 5 3

3 0 1

1 2

a c a

a b b

b c c

24sin 1 ( 3 sin cos )( 3 s inx cos ) 2

3 s inx cos 3 s inx cos

x x x x

x x

Dạng13

( 3 sin cos ) 2( 3 sin cos )

dxI x x dx

x x

3 cos s inx 2( 3 sin cos )

dxx

x x

VÍ DỤ 2 23sin 2sin cos os

dxI

x x x c x

2 2(3 2 1) os

dx

tg x tgx c x

2 2

( ), ( )

3 2 1 os

d tgx dxdtgx

tg x tgx c x

13( 1)( )

3

dtgx

tgx tgx

1( 1) ( )1 3

14 ( 1)( )3

tgx tgxdtgx

tgx tgx

1[ ]

14 13

dtgx dtgx

tgxtgx

1 1[ ln ln 1 ]

4 3tgx tgx C

DẠNG 14

CÁC BÀI TOÁN CHỌN LỌC

ĐHHH99:sin 2 2sin

dxI

x x

1

2sin (cos 1)dx

x x

1

ó : ( s inx,cos ) (s inx,cos )2sin (cos 1)

tac f x f xx x

cos sin xs inx

dtt x dt dx dx

2 2

1 1 1

2s inx( 1) s inx 2 ( 1)(1 ) 2 ( 1) ( 1)

dt dt dtI

t t t t t

2 2

1 1 1 1, ,

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

a b ca b c

t t t t t

2

1 1 1

4 ( 1) 8 1 8 1

dt dt dtI

t t t

1 1 1ln 1 ln 1

4( 1) 8 8t t C

t

os5 sin os(3 2 )s inxcos5

cos cos

c x x c x xxtgx

x x

cos5I xtgxdx

( os3 cos 2 sin 3 sin 2 )s inx

cos

c x x x x

x

3 2 3[( 3cos 4cos )(2cos 1) (3sin 4sin )2sin cos ]s inx

cos

x x x x x x x

x

2 2 2 2[( 3 4cos )(2cos 1) 2s in x(3 4sin )]s inxx x x

2 4 2 2 2[ 6cos 3 8cos 4cos 2(1 os )( 1 4cos )]s inxx x x c x x

ĐHNTTPHCM.KA-2000

2 4 2 2 4[ 6cos 3 8cos 4cos 2 10cos 8cos ]s inxx x x x x

4 2[16cos 20cos 5]s inxx x

4 2(16cos 20cos 5)sin xI x x dx

4 2(16cos 20cos 5) cosx x d x

5 316 20cos os 5cos

5 3x c x x C

CÁCH 2 cos5I xtgxdx

3

2

os5 sin os(3 2 )s inxcos5

cos cos( os3 cos 2 sin 3 sin 2 )s inx

cos

[( 3cos 4cos ) os2 2sin 3 sin cos ]s inx

cos

[( 3 4cos ) os2 2sin 3 sin ]s inx

c x x c x xxtgx

x xc x x x x

x

x x c x x x x

x

x c x x x

2

[[ 3 2(1 os2 )] os2 ( os2 os4 )]s inx

[ 2 os2 2 os 2 os4 ]s inx

[-2cos2x 1 os4 os4 ]s inx

2sin os2 s inx 2sin cos 4

2sin 2sin 3 sin 5

c x c x c x c x

c x c x c x

c x c x

xc x x x

x x x

ĐHQGHN.KA-96 sin( )(2 sin 2 )4

I x x dx

2 sin( ) sin( )sin 24 4

x dx x xdx

1

2 os( ) [ os( ) os(3 )]4 2 4 4

c x c x c x dx

1 12 os( ) sin( ) sin(3 )

4 2 4 6 4c x x x C

4 4

os2

sin os

c xI dx

x c x

2

os2x sin 21 ( 2 sin 2 )( 2 sin 2 )1 sin 22

c d xdx

x xx

( 2 sin 2 ) ( 2 sin 2 )sin 2

( 2 sin 2 )( 2 sin 2 )

x xd x

x x

sin 2 sin 2

2 sin 2 2 sin 2

d x d x

x x

ln 2 sin 2 ln 2 sin 2x x C

VÍ DỤ

VÍ DỤ3 5sin cos

dxI

x x

4 6

sin x cos

sin cos

xdx

x x

2

2 2 6

1 (cos )

2 (1 os ) os

d x

c x c x

2 2os cosc x t d x dt 2 3

1

2 (1 )

dtI

t t

3

2 3 2 3

1 [ (1 )]

(1 ) (1 )

t t

t t t t

3 2 2 3

2 3

3 (1 ) 3 (1 ) (1 )

(1 )

t t t t t t

t t

2 3

1 3 3 1

(1 ) (1 )

t

t t t t t

2

1

(1 )t

1 1

3( )1t t

3 2

3 1 1

t t t

2 3 2

1 6 3 1 1

(1 ) 1t t t t t

2 3 2

1 3 1 13

2 (1 ) 2 1 2 2

dt dt dt dt dtI

t t t t t

2

1 3 1 13ln ln 1

2(1 ) 2 4 2t t C

t t t

VÍ DỤ 2

sin 2

1 sin

xdxI

x

2

2sin cos

1 sin

x xdx

x

2

2

(sin )

1 sin

d x

x

2ar (sin )ctg x C

Nhận xét

2

2

2 2

sin 2 sin cos sin 2

cos 2 sin cos sin 2

( sin cos ) ( )sin 2

da x a x xdx a xdx

db x b x xdx b xdx

d a x b x a b xdx

VÍ DỤ:4

3

os

sin

c xdxI

x

4 4

4 2 2

os os (cos )sin xdx=-

sin (1 os )

c x c xd x

x c x

cos cosx t d x dt 4 4

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

t dt t dtI

t t t

4

2 2(1 ) (1 )

t

t t

2

2 2

[(1 ) (1 )]

(1 ) (1 )

t t

t t

( s inx)= (s inx)f f Ta có:

2 2

1 2 1

(1 ) (1 )(1 ) (1 )t t t t

2 22

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

dt dt dtI

t t t t

1 1 1ln

1 1 1

tC

t t t

VÍ DỤ2 2

sin 2

2sin 3cos

xdxI

x x

2 2 2 2 22sin 3cos 2sin 3cost x x x x t

2 2(2sin 3cos ) 2d x x tdt

(4sin cos 6cos sin ) 2x x x x dx tdt

2sin cos 2 sin 2 2x xdx tdt xdx tdt

2 222 2 2sin 3cos

tdtI t C x x C

t

VÍ DỤ 2

sin 4

1 os

xI dx

c x

2

2sin 2 cos 2

1 os

x xdx

c x

2ó : 2sin cos ( os )tac x xdx d c x

2 2

2

2(2cos 1) ( os )

1 os

x d c xI

c x

2 2

2

[2( os 1) 3] ( os )2

1 os

c x d c x

c x

2

2

4sin cos (2cos 1)

1 os

x x x dx

c x

22

2

( os )4 ( os ) 6

1 os

d c xd c x

c x

2 2

24cos 6 ( os ) , ( ar )

1

dux arctg c x C ctgu C

u

cos s inx

3 sin 2

xI dx

x

ó : (cos s inx) (s inx cos )tac x dx d x

23 sin 2 4 (s inx cos )x x

2

(cos s inx)

4 (s inx cos )

x dxI

x

2

(s inx cos )

4 (s inx cos )

d x

x

s inx cost x 24

dtI

t

VÍ DỤ

2sin 2cos , ( )2 2

t y dt ydy y

2cosarcsin( )

2cos 2

ydy tI y C

y

s inx cosarcsin( )

2

xC

2 2arcsin

du uC

aa u

Cần nhớ

ĐHTMHN-20002

30

4sin x

(s inx cos )

dxI

x

2

30

4sin

2 2 os ( )4

xdx

c x

1s inx sin( ) (sin cos )

4 4 2x t t t t

2

30

2(sin cos )

2. 2 os

t tI dt

c t

2

30

sin cos

os

t tdt

c t

2 2

3 20 0

sin

os os

tdt dt

c t c t

4 4

3 2 2

4 4

(cos ) 1( )

os os 2 os

d t dttgt

c t c t c t

4

4

2

'4sin (s inx cos ) (s inx cos )x a x b x

'

3 32

4sin 2 2(s inx cos )

(s inx cos ) (s inx cos )2sin ( )4

x x

x xx

Cách 2:

C3:2

30

4sin x

(s inx cos )

dxI

x

2

30

4cos

(s inx cos )

xdxJ

x

2

20

4ét :

(s inx cos )

dxx I J

x

2

20

4sin ( )

4

dx

x

2

30

(cos s inx)ét : 0

(s inx cos )

x dxx I J

x

2I J

ĐHTL

2 2

3sin

3sin 4cos

xdx

x x

2 2

4cos

3sin 4cos

xdx

x x

2 2

3s inx 4cos

3sin 4cos

xI dx

x x

2

(cos )3

3 os

d x

c x

2

(s inx)4

4 sin

d

x

1 cos3 ar

3 3

xctg

2 s inxln

2 s inxC

Cần nhớ

2 2

1ar

1ln

( )( ) 2

du uctg c

u a a adx a x

a x a x a a x

ĐHNNHNos2

1 cos

c xdxI

x

2

2

1 2sin

2sin2

xdx

x

22sin2

dxx

2

2

sin

sin2

xdxx

2 2

2

4sin os2 2cot

2 sin2

x xcx

dxx

cot 2 (1 cos )2

xx dx cot 2 2sin

2

xx x C

HVKTQS3

4

4sin

1 os

xI dx

c x

2

4

4sin sin

1 os

x xdx

c x

2

4

4sin(cos )

1 os

xd x

c x

2

4

1 os4 (cos )

1 os

c xd x

c x

cos x t

2

4

14

1

tI dt

t

2

22

11

1t dtt

t

2

1( )

1( ) 2

d tt

tt

121

ln12 2 2

tt Ct

t

2

2

1 os 2 cos 1ln

2 2 os 2 cos 1

c x xC

c x x

VÍ DỤsin cos 1

dxI

x x

22sin os 2cos 1 12 2 2

dxx x xc

2

1

2 ( 1) os2 2

dxx x

tg c

2

ó : ( )2 2 os

2

x dxtac d tg

xc

2 ( )1 2 ln 12 21

2

xd tg x

I tg Cx

tg

4

sin 2

1 sin

xdxI

x

2

2 2

(sin )

(sin ) 1

d x

x

2ar (sin )ctg x C

4

sin 2

1 os

xdxI

c x

VÍ DỤ

22

2 2

( os )ar ( os )

( os ) 1

d c xctg c x C

c x

HVKTQS3 3

3

sin s inxotx

sin

xI c dx

x

3 3

2 2

sin s inx cos

sin s in x

x xdx

x

2

1 cos

s inx sin

xt dx dt

x

33

2

1 1

1t tI dt

t

32 2 23

3

1(1 ) 1t t dt t tdt

t

3 2 2 2 431 31 (1 ) (1 )

2 8t d t t C

3 3

2 2

sin s inx sin

sin s in x

x d x

x

VÍ DỤ4

os2

tg xdxI

c x

2 22

(1 ) (1 )os

dxt tgx dt tg x dx t dx

c x

2

2

1os2

1

tc x

t

4

2 2

2

1 (1 )1

t dtI

t tt

42

2 2

1( 1 )

1 1

tdt t dt

t t

3 1 1

ln3 2 1

t tt C

t

3

2

4

cos os 1

tgxdxI

x c x

2 2

1 sin x cos

cos os os

dx d xt dt

x c x c x

2 2

22 2

2

1 11

dt tdtI

tt

3

2 2

4

sin x

os os 1

dx

c x c x

3

2 2

4

(cos )

os os 1

d x

c x c x

VÍ DỤ

2 12 2 22

2

1( 1) ( 1) 1

2t d t t

5 3

2

2

VÍ DỤ32

4 20

os

os 3cos 3

c xdxI

c x x

22

2 2 20

(1 sin ) sin

(1 sin ) 3(1 sin ) 3

x d x

x x

22

4 20

1 sinsin

sin sin 1

xd x

x x

s inx t

1 2

4 20

1

1

tI dt

t t

1 2

202

11

11

t dtt

t

1

20

1( )

1( ) 1

d tt

tt

1

0

1( )

1 1( 1)( 1)

d tt

t tt t

111

ln12 1

tt

tt

0

12

2

1 1ln

2 1

t t

t t

1

ln 32

VÍ DỤ4

0 1

dxJ

tgx

4

0

cos

s inx cos

xdxx

'1 1(s inx cos ) (s inx cos )cos 2 2

s inx cos s inx cos

x xx

x x

'4 4

0 0

1 1 (s inx cos )

2 2 (s inx cos )

x dxJ dx

x

40

1 2ln s inx cos

8 2 8 2x

4'

0

sin x

s inx cos

dxJ

x

'

'

4

2

J J

J J

2

8 2J

VÍ DỤ4

20 ( 1)

dxK

tgx

24

20

os

(s inx cos )

c xdx

x

24

20

sin

(s inx cos )

xL dx

x

4

20

ó :(s inx cos )

dxtac K L

x

440

20

1 1cot( )

2 4 22sin ( )4

dxx

x

2 24

20

( os sin )ó :

(s inx cos )

c x x dxtac K L

x

4

0

cos s inx

s inx cos

xdxx

4

40

0

(s inx cos )ln s inx cos 2

s inx cos

d xx

x

1 2

4 2K

VÍ DỤ4

2

0

1( )

1

tgxI dx

tgx

24

20

[( 1) 2]

( 1)

tgxdx

tgx

24

20

( 1) 4( 1) 4

( 1)

tgx tgxdx

tgx

4 4 4

20 0 0

4 41 ( 1)

dx dxdx

tgx tgx

40/ 4 4 2 2 1 2 2 1

4 2 4x J K

VÍ DỤ2

2

3

cos

(1 cos )

xdxI dx

x

22

4

3

1 2sin2

4sin2

x

dxx

2 2

4 2

3 3

1 1

4 2sin sin2 2

dx dxx x

22 2

33

1(cot 1)2 (cot ) cot /

4 2 2 2

x x xd

3 2 2

3 3

1[cot cot ] cot /

2 2 2 2

x x x

3

3os

cos 1

c xdxI

x

3( os 1) 1

cos 1

c xdx

x

2( os cos 1)cos 1

dxc x x dx

x

2

1 os2cos

2 2cos2

c x dxdx xdx dx

x

2

21 1 2os22 2 2 cos

2

xddx

c xdx xx

1sin 2

2 4 2

x xx x tg C

VÍ DỤ

VÍ DỤ 4sin cos

dxI

x x 4 2

cos

sin cos

xdx

x x

s inx cost dt xdx

4 2(1 )

dtI

t t

2 2 2( ) (1 )

dt

t t

2 2 2

2 2 2

[ (1 )]

( ) (1 )

t tdx

t t

4 2 2 2 2

4 2

2 (1 ) (1 )

(1 )

t t t tdt

t t

2

2 2 4

12

1

dt dt tdt

t t t

2 2 4 22

1

dt dt dt dt

t t t t

3

1 1 1 1ln

2 1 3

tC

t t t

Ví dụ2 2

23 3

s inx 1 cos2sin os 2cos

2 2 2

dx dxI

x x x xc

2

2

3

1

2 2 sin os2 2

dxx xc

2

2 2

3

sin1 22 2 (1 os ) os

2 2

xdx

x xc c

2

2 2

3

2 ( os )1 22 2 (1 os ) os

2 2

xd c

x xc c

2 22

2 2

3

1 os os1 2 2 ( os )22 (1 os ) os

2 2

x xc c x

d cx x

c c

2 2

2 2

3 3

( os ) ( os1 2 2[ ]2 os (1 os )

2 2

x xd c d c

x xc c

22

3 3

1 os1 1 1 2ln222 os 1 os

2 2

xc

x xc c

1 2 2 1 2 2 2 3( ) [ ln( ) ln( )]

2 2 3 2 2 2 2 2 3

Ví dụ2

3

2

cos cos osI x x c xdx

22

2

cos sin cosx x xdx

2

2

cos s inx cosx xdx

0

2

cos sin x cosx xdx

2

0

cos sin x cosx xdx

122

0

2 (cos ) cos cosx xd x

3 52 22 2

0 0

1 12 (cos ) cos cos

5 5x d x x

Ví dụ2

2

0 0

1sin cos sin (1 os2 )

2I x x xdx x x c x dx

0 0

1 1sin sin cos 2

2 2x xdx x x xdx

0 0

1 1sin (sin 3 s inx)

2 4x xdx x x dx

0 0

1 1sin sin 3

4 4x xdx x xdx

0 0

1 1cos cos3

4 12xd x xd x

udv uv vdu

0 00 0

1 1( cos cos ) ( cos3 os3 )

4 12x x xdx x x c xdx

1 1( 0) ( 0)

4 12 3

Cách 2:

0

( )m

I f x dx x m t

2

0

sin cosI x x xdx

,s inx sin ,cos cosx t dx dt t x t 0

2( )sin cosI t t tdt

2 2 2

0 0 0

sin cos sin cos os cost tdt t t tdt c td t I

32

0 0

os 1 1 22 os cos ( )

3 3 3 3 3

c tI c td t I

VÍ DỤ

VÍ DỤ0

sin 4

1 s inx

xI dx

0

2sin 2 cos 2

1 s inx

x xdx

2

0

4sin cos (1 2sin )

1 s inx

x x x dx

2

0

4sin (1 2sin ) sin

1 s inx

x x d x

3

0

(4sin 8sin ) sin

1 s inx

x x d x

1 s inx sint d x dt

1 3

1

[4( 1) 8( 1) ]dt0

t tI

t

C2: , 4 4 4x t dx dt x t

sin 4 sin 4 ,s inx sinx t t

0

0

sin 4 sin 4

1 sin 1 sin

tdt tdtI I

t t

2 0 0I I

VÍ DỤ2

4

4

sin

dxI

x

2 4

4

4

osdxc xtg x

24 2

4

( )tg x tg x dtgx

3 1 2

4

4( )

3 1 3

tg tg

22 22 2

4 4

4 4

1(1 )os os

dxtg x dtgxc x c x

tg x tg x

C2:

VÍ DỤ 22

sin 3

0

sin x cosxI e xdx

22

sin 2

0

(1 sin )sin x cosxe x xdx

22

sin 2 2

0

1(1 sin ) (sin )

2xe x d x

2 22

sin 2 sin 2 2

0 0

1 1(sin ) sin sin

2 2x xe d x e xd x

2 222

sin sin 2 2

0 0

1 1sin sin

2 2x xe e xd x

22

sin 2 2

0

1 1 1sin sin

2 2 2xe e xd x

22

sin 2 2

0

1sin sin

2xK e xd x

2sin x t

1

0

1

2tK e tdt

1

0

1

2ttde

22

sin 2 2

0

1 1 1sin sin

2 2 2xe e xd x

11 1

0 00

1 1 1( ) ( )

2 2 2t t tte e dt e e

1 1 11

2 2 2 2

eI e

udv uv vdu

VÍ DỤ3

0

s inxln(cos )I x dx

3

0

ln cos (cos )xd x

30

cos ln cosx x

3

0

s inxcos .

cosx dx

x

3

30

0

1 1 1 1ln sin x ln cos

2 2 2 2dx x

1 1 1ln 1

2 2 2

VÍ DỤ 6

0

sin2

xI dx

6

0

os2

xJ c dx

2

0 0 0

1 1(1 sin ) (1 os2 )

2 4I J x dx dx c x dx

0 0

3 1 3sin 2

4 8 4x x

,2 2 2

x tx t dx dt

06 6

0

sin sin2 2

t tJ dt dt I

3

8I J

2 2

0 0

(s inx) (cos )f dx f x dx

VÍ DỤ42

4 40

sin

os os

xdxI

c x c x

42

4 40

osét :

sin os

c xdxx J

x c x

:cm I J ,s inx cos2

x t dx dt t

0 4

4 4

2

os

sin os

c tI dt

t c t

42

4 40

os

sin os

c tdt J

t c t

2

20

0

ính: I+J=2

tat dx x

4I J

2

0

s inx

s inx cos

dxI dx

x

2

0

cos

s inx cos

xdxJ dx

x

ó :4

2

I Jtac I J

I J

VÍ DỤ

VD:2

3 3

0

( cos sin )I x x dx

2 23 3

0 0

cos sinxdx xdx

2 2

3 3

0 0

cos & sinK xdx L xdx

, cos sin2

x t dx dt x t

0 23 3

02

sin sinK tdt tdt L

0I K L