chuyen-de-nguyen-ham-tich-phan.pdf

LUYN THI I HC MN TON Thy Hng Chuyn Nguyn hm Tch phn

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I. NHC LI KHI NIM V VI PHN CA HM S

Vi phn ca hm s y = f(x) c k hiu l dy v cho bi cng thc = = =( ) ' '( )dy df x y dx f x dx V d: d(x2 2x + 2) = (x2 2x + 2)dx = (2x 2)dx d(sinx + 2cosx) = (sinx + 2cosx)dx = (cosx 2sinx)dx Ch : T cng thc vi phn trn ta d dng thu c mt s kt qu sau ( ) ( )12 2 2

2d x dx dx d x= =

( ) ( )13 3 33

d x dx dx d x= =

( ) ( ) ( )2 2 2 21 1 12 2 2 2xxdx d d x d x a d a x = = = = ( ) ( ) ( )32 3 3 31 1 13 3 3 3xx dx d d x d x a d a x = = = =

( ) ( ) ( )ax1 1 ln ax lnax

d bdx dxd b d xax b a b a x

+= = + =

+ +

( ) ( ) ( ) ( )( ) ( )1 1 1sin ax sin ax ax cos ax sin 2 os2 ...2

b dx b d b d b xdx d c xa a

+ = + + = + =

( ) ( ) ( ) ( )( ) ( )1 1 1cos cos sin cos2 sin 2 ...2

ax b dx ax b d ax b d ax b xdx d xa a

+ = + + = + =

( ) ( ) ( )ax 2 21 1 1ax ...2b ax b ax b x xe dx e d b d e e dx d ea a+ + += + = = ( )

( )( ) ( ) ( )2 2 2ax1 1 1

tan tan 2 ...2cos cos cos 2

d bdx dxd ax b d xa aax b ax b x

+= = + =

+ +

( )( )

( ) ( ) ( )2 2 2ax1 1 1

cot cot 2 ...2sin sin sin 2

d bdx dxd ax b d xa aax b ax b x

+= = + =

+ +

II. KHI NIM V NGUYN HM

Cho hm s f(x) lin tc trn mt khong (a; b). Hm F(x) c gi l nguyn hm ca hm s f(x) nu F(x) = f(x) v c vit l ( )f x dx . T ta c : ( ) ( )f x dx F x= Nhn xt: Vi C l mt hng s no th ta lun c (F(x) + C) = F(x) nn tng qut ha ta vit ( ) ( )f x dx F x C= + , khi F(x) + C c gi l mt h nguyn hm ca hm s f(x). Vi mt gi tr c th ca C th ta c mt nguyn hm ca hm s cho. V d: Hm s f(x) = 2x c nguyn hm l F(x) = x2 + C, v (x2 + C) = 2x Hm s f(x) = sinx c nguyn hm l F(x) = cosx + C, v (cosx + C) = sinx III. CC TNH CHT C BN CA NGUYN HM

Cho cc hm s f(x) v g(x) lin tc v tn ti cc nguyn hm tng ng F(x) v G(x), khi ta c cc tnh cht sau:

a) Tnh cht 1: ( )( ) ( )f x dx f x = Chng minh:

Ti liu tham kho:

01. M U V NGUYN HM Thy ng Vit Hng

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Do F(x) l nguyn hm ca hm s f(x) nn hin nhin ta c ( ) ( )( ) ( ) ( )f x dx F x f x = = pcm. b) Tnh cht 2: [ ]( )( ) ( ) ( ) ( )f x g x dx f x dx g x dx+ = + Chng minh:

Theo tnh cht 1 ta c, ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )f x dx g x dx f x dx g x dx f x g x + = + = + Theo nh ngha nguyn hm th v phi chnh l nguyn hm ca f(x) + g(x). T ta c [ ]( )( ) ( ) ( ) ( )f x g x dx f x dx g x dx+ = + c) Tnh cht 3: ( ). ( ) ( ) , 0k f x dx k f x dx k= Chng minh:

Tng t nh tnh cht 2, ta xt ( )( ) . ( ) . ( ) ( )k f x dx k f x k f x dx k f x dx = = pcm. d) Tnh cht 4: ( ) ( ) ( ) ..f x dx f t dt f u du= = Tnh cht trn c gi l tnh bt bin ca nguyn hm, tc l nguyn hm ca mt hm s ch ph thuc vo hm, m khng ph thuc vo bin. IV. CC CNG THC NGUYN HM

Cng thc 1: dx x C= +

Chng minh: Tht vy, do ( ) 1x C dx x C+ = = + Ch : M rng vi hm s hp ( )u u x= , ta c du u C= +

Cng thc 2:n 1

n xx dx Cn 1

+

= ++

Chng minh:

Tht vy, do 1 1

1 1

n nn nx xC x x dx C

n n

+ + + = = +

+ +

Ch : + M rng vi hm s hp ( )u u x= , ta c

1

1

nn uu du C

n

+

= ++

+ Vi 1 2 2 22 2

dx dx dun x C u C

x x u= = = + = +

+ Vi 2 21 12 dx dun C C

x x u u= = + = +

V d:

a) 3

2

3x

x dx C= +

b) ( ) 54 4 22 2 5x

x x dx x dx xdx x C+ = + = + +

c) 1 1

22 2 2 23 3 333 312 2 2

3

x x x x x x xdx dx xdx x dx C x Cx x

= = = + = +

d) ( ) ( ) ( ) ( )5

4 4 2 112 1 2 1 2 12 5

nu du xI x dx x d x I C+

= + = + + = +

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e) ( ) ( ) ( ) ( )2011

2010 2010 1 311 3 1 3 1 33 2011

nu du xI x dx x d x I C

= = = +

f) ( )( )

( ) ( )2

2 2

2 11 1 1 1.

2 2 2 1 2 2 12 1 2 1

duu

d xdxI I C Cx xx x

+= = = + = +

+ ++ +

g) ( ) ( ) ( )3 32 21 1 2 34 5 4 5 4 5 . 4 5 4 54 4 3 8

I x dx x d x I x C x C= + = + + = + + = + +

Cng thc 3: lndx x Cx

= +

Chng minh:

Tht vy, do ( ) 1ln lndxx C x Cx x

+ = = + Ch : + M rng vi hm s hp ( )u u x= , ta c lndu u C

u= +

+ ( )

1 ln 21 1 2x 2ln ax1ax ax ln 2

2 2

dxx k Cd ax bdx kb C

dxb a b a k x Ck x

= + ++ +

= = + + + +

= +

V d:

a) 4

3 31 1 1 2 ln4

dx xx dx x dx dx x x C

x xx x

+ + = + + = + + +

b) ( )3 21 1 ln 3 23 2 3 3 2 3

duu

d xdxI I x Cx x

+= = = + +

+ +

c) ( )2 2 22 12 3 3 3 32 2 3 ln 2 12 1 2 1 2 1 2 2 1 2

d xx x dxdx x dx xdx x x x Cx x x x

++ + = + = + = + = + + + + + + +

Cng thc 4: sinx cosdx x C= +

Chng minh: Tht vy, do ( )cos sin x sinx cosx C dx x C + = = + Ch : + M rng vi hm s hp ( )u u x= , ta c sinu cosdu u C= +

+ ( ) ( ) ( ) ( )1 1 1sin ax sin ax ax cos ax sin 2 os22

b dx b d b b C xdx c x Ca a

+ = + + = + + = + V d:

a) ( )32 2 11 1sinx sinx cos2 1 2 1 2 2 1

d xdxx x dx x xdx dx x dx x

x x x

+ + = + + = + =

522 1

cos ln 2 15 2x

x x C= + +

b) ( ) ( )4 33 1 3 1 3sin 2 sin 2 3 sin 2 2 os2 ln 4 34 3 4 3 2 4 4 3 2 4

d xdxx dx xdx xd x c x x C

x x x

+ = + = + = + +

c) sin sinx sin32x

x dx + +

Ta c ( ) ( ) ( ) ( )1 1 12 ; 2 2 2 ; 3 3 32 2 2 2 3x xd dx dx d d x dx dx d x d x dx dx d x = = = = = =

T :

( ) ( )1 1sin sinx sin3 sin sin 2 sin3 2 sin sin 2 2 sin3 32 2 2 2 2 3x x x x

x dx dx xdx xdx d xd x xd x + + = + + = + +

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1 12cos os2 os32 2 3x

c x c x C= +

Cng thc 5: cos sinxdx x C= +

Chng minh: Tht vy, do ( )in cos cosx inxs x C x dx s C+ = = + Ch : + M rng vi hm s hp ( )u u x= , ta c cosu sindu u C= +

+ ( ) ( ) ( ) ( )1 1 1os ax os ax ax sin ax os2 sin 22

c b dx c b d b b C c xdx x Ca a

+ = + + = + + = + V d:

a) 4 1 5cos sin cos sin x 4 sinx cos 4 5ln 11 1

xx x dx xdx dx dx x x x C

x x

+ = + = + + + + + +

b) ( )21

cos 2 sin x os2 sinx sin 2 cos2 2

xx x dx c xdx dx xdx x x C+ = + = +

c) ( )2 1 os2 1 1 1 1 1 1sin os2 os2 2 sin 22 2 2 2 4 2 4

c xxdx dx c x dx x c xd x x x C = = = = +

Cng thc 6: 2 tancosdx

x Cx

= +

Chng minh:

Tht vy, do ( ) 2 21tan tan xcos cosdx

x C Cx x

+ = = + Ch : + M rng vi hm s hp ( )u u x= , ta c 2 tan uos

du Cc u

= +

+ ( )( )

( ) ( )2 2 21 1 1

tan tan 2cos cos cos 2 2

d ax bdx dxax b C x C

ax b a ax b a x+

= = + + = ++ +

V d:

a) 2 21 1

cos sin 2 cos sin 2 tan sin cos2cos cos 2

dxx x dx xdx xdx x x x C

x x

+ = + = + + +

b) ( ) ( )( )

( )( )

2 2 2

2 1 5 41 2 1 22cos 2 1 5 4 cos 2 1 5 4 2 cos 2 1 4 5 4

d x d xdx dxI dxx x x x x x

= + = + =

( )2os 1 1tan 2 1 ln 5 42 2

duc u x x C = +

c) ( )( )

( ) ( )2os

2 2

3 21 1tan 3 2

cos 3 2 2 cos 3 2 2

duc u

d xdxI I x Cx x

= = = +

Cng thc 7: 2 cot xsindx C

x= +

Chng minh:

Tht vy, do ( ) 2 21cot cot xsindx

x C Csin x x

+ = = +

Ch : + M rng vi hm s hp ( )u u x= , ta c 2 cot usin

du Cu

= +

+ ( )( )

( ) ( )2 2 2ax1 1 1

cot ax cot 2sin ax sin ax sin 2 2

d bdx dxb C x Cb a b a x

+= = + + = +

+ +

V d:

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a) 6

5 52 2

1 1cos 2 2 cos 2 2 sin 2 cot

sin sin 2 3dx x

x x dx xdx x dx x x Cx x

+ = + = + + +

b) ( )( )

( ) ( ) ( )2sin

2 2

1 31 1 1cot 1 3 cot 1 3

sin 1 3 3 sin 1 3 3 3

duu

d xdxI I x C x Cx x

= = = + = +

c) 2sin2 2

22 2cot2

sin sin2 2

duu

xddx xI I C

x x

= = = +

Cng thc 8: x xe dx e C= +

Chng minh:

Tht vy, do ( )x x x xe C e e dx e C+ = = + Ch : + M rng vi hm s hp ( )u u x= , ta c u ue du e C= +

+ ( )2 2

2 2

11 1 2

ax12

x k x k

ax b ax b ax b

k x k x

e dx e Ce dx e d b e C

a ae dx e C

+ +

+ + +

= +

= + = +

= +

V d:

a) ( ) ( )2 1 2 1 2 12 2 231 4 4 1 12 1 4.2sin 3 sin 3 2 3 sin 3x x x

d xdxe dx e dx dx e d x x

x x xx x

+ + + + = + = + +

2 11 1 cot 3 82 3

xe x x C += + + +

b) ( )( ) ( ) ( ) ( ) ( )3 2 3 2 3 24 14 os 1 3 4 os 1 3 3 2 os 1 3 1 33 3x x xe c x dx e dx c x dx e d x c x d x+ + ++ = + = + ( )3 24 1 sin 1 3

3 3xe x C+= +

Cng thc 9:ln

xx aa dx C

a= +

Chng minh:

Tht vy, do lnln ln ln

x x xx xa a a aC a a dx C

a a a

+ = = = +

Ch : + M rng vi hm s hp ( )u u x= , ta c u ua du a C= +

+ ( )1 1kx m kx m kx ma dx a d kx m a Ck k

+ + += + = +

V d:

a) ( ) ( ) ( ) 3 23 2 3 2 3 21 1 2 32 3 2 3 2 3 3 23 2 3ln 2 2ln3u

x xa dux x x x x xI dx dx dx d x d x I C= + = + = + = + +

b) ( ) ( ) ( ) 1 21 2 4 3 1 2 4 3 1 2 4 3 4 31 3 2 32 2 3 2 1 2 4 32 4 2ln 2 4x

x x x x x x xe dx dx e dx d x e d x e C

+ + + + = = + = + +

BI TP LUYN TP: 1) ( )51 2I x x dx= + 2) 3 52 71 3I x dxx = 3) ( )5 2 3 33 4 2I x x x dx= +

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4) 34 251 24 xI x dx

xx

= +

5) 5

1x + dx

xI =

6)

4

6 22 3xI dx

x

+=

7) ( )2

7

1xI dx

x

= 8) ( )238 2 1I x dx= 9) ( )22

9 2

4xI dx

x

+=

10) 4 3 2

10 23 2 1x x xI dx

x

+ += 11)

2

11x x x xI dx

x

= 12) 12 31 1I dxx x

=

13) 3

131I x dxx

=

14)

2

14 3

1I x dxx

= +

15)

( )2315

2 3x xI dx

x

=

16) ( )( )416 2I x x x x dx= 17) 17 51(2 3)I dxx= 18) 18 41( 3)xI dxx += 19) 19

pisin

2 7xI dx = +

20) 20 sin 2 sin 3

xI x dx = +

21) 21 sin 2xI x dx = +

22) 22pi 1

sin 3 sin4 2

xI x dx + = +

23) 223 cos 2xI dx= 24) 224 sin 2

xI dx=

26) 26 2cos 4dxI

x= 27) ( )27 2cos 2 1

dxIx

=

28) ( )228 tan 2I x x dx= + 29) 429 tanI x dx= 30) 230 cotI x dx= 31) ( )31 2sin 2 3

dxIx

=

+

32) 32 1 cos6dxI

x=

33) 2 233 21

cot dxI x xx

= + +

34) 234

1 dx3 2

I xx

= + +

35) 2351

sin2 5

I x dxx

=

36) 36

2 dx3

xIx

+=

37) 372 14 3

xI dxx

=

+

38) 38 6 5xI dx

x=

39) 2

3911

3x xI dx

x

+ +=

+ 40)

2

402 5

1x xI dx

x

+=

41) 3 2

413 2 1

2x x xI dx

x

+ + +=

+ 42)

3 2

424 4 1

2 1x xI dx

x

+ =

+ 43)

2

434 6 1

2 1x xI dx

x

+ +=

+

44) 2x 344I e dx += 45) 3 145 cos(1 ) xI x e dx = + 46) 2 1

46 .xI x e dx +=

47) 47 22

sin (3 1)xI e dx

x

= +

+ 48) 48 22 cos

xx eI e dx

x

= +

49) ( )1 2 4 349 2 x xI e dx +=

50) 5012x

I dx= 51) 5127

x

xI dx= 52) 2 152 3 xI dx+=

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1. Khi nim nguyn hm Cho hm s f xc nh trn K. Hm s F c gi l nguyn hm ca f trn K nu '( ) ( )F x f x= , x K Nu F(x) l mt nguyn hm ca f(x) trn K th h nguyn hm ca f(x) l ( ) ( )f x dx F x C= + , C R.

Mi hm s f(x) lin tc trn K u c nguyn hm trn K. 2. Tnh cht

'( ) ( )f x dx f x C= +

[ ]( ) ( ) ( ) ( )f x g x dx f x dx g x dx = ( ) ( ) ( 0)kf x dx k f x dx k=

3. Bng nguyn hm ca mt s hm s thng gp

0dx C=

dx x C= +

1, ( 1)

1x

x dx C+

= + +

1

lndx x Cx

= +

x xe dx e C= +

(0 1)ln

xx aa dx C a

a= + <

cos sinxdx x C= +

sin cosxdx x C= +

21

tancos

dx x Cx

= +

21

cotsin

dx x Cx

= +

1

cos( ) sin( ) ( 0)ax b dx ax b C aa

+ = + +

1

sin( ) cos( ) ( 0)ax b dx ax b C aa

+ = + +

1

, ( 0)ax b ax be dx e C aa

+ += +

1 1

lndx ax b Cax b a

= + ++

V d 1. Chng minh F(x) l mt nguyn hm ca hm s f(x) bit rng

a) ( ) (4 5)( ) (4 1)

x

x

F x x ef x x e

=

= b)

4

5 3( ) tan 3 5( ) 4 tan 4 tan 3

F x x xf x x x

= +

= + +

c)

2

2

2 2

4( ) ln3

2( )( 4)( 3)

xF xx

xf xx x

+=

+ =

+ +

d)

2

2

2

4

2 1( ) ln2 1

2 2( 1)( )1

x xF xx x

xf xx

+

= + +

=

+

Ti liu bi ging:

01. M U V NGUYN HM Thy ng Vit Hng

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V d 2. Tm cc nguyn hm sau

1) 2 1 3 ..........................................................................x x dxx

+ =

2)4

22 3

..................................................................................

x dxx

+=

3) 21

...................................................................................

x dxx

=

4)2 2

2( 1)

..............................................................................

x dxx

=

5) ( )3 4 ......................................................................................x x x dx+ + = 6)

3

1 2...............................................................................dx

x x

=

7) 22sin .............................................................2x dx =

8) 2tan ............................................................................xdx =

9) 2cos ................................................................xdx =

10) 2 21

.........................................................................................

sin .cosdx

x x=

11) 2 2cos 2

....................................................................................................................................

sin .cosx dx

x x=

12) 2sin 3 cos 2 ............................................................................................x xdx =

13) ( ) 1 .............................................................................x xe e dx =

14) 22 .......................................................................................cosx

x ee dxx

+ =

15) 3 1 2 ......................................................................................................................1

x xe dxx

+ + =

V d 3. Tm nguyn hm F(x) ca hm s f(x) tho iu kin cho trc: a) 3( ) 4 5; (1) 3f x x x F= + = b) pi= =( ) 3 5cos ; ( ) 2f x x F

c) 23 5

( ) ; ( ) 1x

f x F ex

= = d) 2 1 3

( ) ; (1)2

xf x F

x

+= =

e) = =3

21( ) ; ( 2) 0xf x F

x f) 1( ) ; (1) 2f x x x F

x= + =

g) pi= =

( ) sin2 .cos ; ' 03

f x x x F h) 4 3

23 2 5( ) ; (1) 2x xf x F

x

+= =

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i) 3 3

23 3 7

( ) ; (0) 8( 1)

x x xf x F

x

+ + = =

+ k) 2 pi pi( ) sin ;

2 2 4xf x F = =

V d 4. Cho hm s g(x). Tm nguyn hm F(x) ca hm s f(x) tho iu kin cho trc:

a) pi = + = =

2( ) cos ; ( ) sin ; 32

g x x x x f x x x F

b) pi= + = =2( ) sin ; ( ) cos ; ( ) 0g x x x x f x x x F

c) 2( ) ln ; ( ) ln ; (2) 2g x x x x f x x F= + = = V d 5. Tm iu kin ca tham s hm s F(x) l mt nguyn hm ca hm s f(x):

a) 3 2

2( ) (3 2) 4 3. .( ) 3 10 4F x mx m x x

Tm mf x x x

= + + +

= + b)

2

2

( ) ln 5. .2 3

( )3 5

F x x mxTm mx

f xx x

= +

+=

+ +

c) 2 2

2

( ) ( ) 4 . , , .( ) ( 2) 4

F x ax bx c x xTm a b c

f x x x x

= + +

=

d) 2( ) ( ) . , , .

( ) ( 3)

x

x

F x ax bx c eTm a b c

f x x e

= + +

=

e) 2 2

2 2( ) ( ) . , , .( ) (2 8 7)

x

x


f x x x e

= + +

= + f)

2

2( ) ( ) . , , .( ) ( 3 2)

x

x


f x x x e

= + +

= +

g) ( ) ( 1)sin sin2 sin3 . , , .2 3( ) cos

b cF x a x x x Tm a b c

f x x

= + + + =

h) 2

2( ) ( ) 2 3

. , , .20 30 7( )

2 3

F x ax bx c x

Tm a b cx xf x

x

= + +

+

=

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CC BIU THC VI PHN QUAN TRNG

1. ( ) ( ) ( )2 2 21 1 12 2 2xdx d x d x a d a x= = = 6. ( ) ( ) ( )2 cot cot cotsindx d x d x a d a x

x= = =

2. ( ) ( ) ( )2 3 3 31 1 13 3 3x dx d x d x a d a x= = = 7. ( ) ( ) ( )2dx d x d x a d a xx = = = 3. sin (cos ) (cos ) ( cos )x dx d x d x a d a x= = = 8. ( ) ( ) ( )x x x xe dx d e d e a d a e= = = 4. cos (sin ) (sin ) ( sin )x dx d x d x a d a x= = = 9. ( ) ( ) ( )ln ln lndx d x d x a d a x

x= = =

5. ( ) ( ) ( )2 tan tan tancosdx d x d x a d a x

x= = = 10. ( ) ( )1 1dx d ax b d b ax

a a= + =

V d 1. Tm nguyn hm ca cc hm s sau:

a) 1 21xI dxx

=

+ b) 2 10

2 (1 )I x x dx= + c) 2

3 3 1x dxIx

=

+

Hng dn gii:

a) S dng cc cng thc vi phn ( ) ( )

( )

22 21 1

2 2 2

ln

xxdx d d x d x a

du d uu

= = =

=

Ta c ( ) ( ) ( )

2 2(ln ) ln 2

1 12 2 2

11 1 1 ln 1 .2 2 21 1 1

du d u u Cu

d x d xxI dx I x Cx x x

= = ++= = = = + +

+ + +

b) S dng cc cng thc vi phn ( ) ( )2 2 2

1

1 12 2 2

1

nn

xxdx d d x d x a

uu du d

n

+

= = =

= +

Ta c ( ) ( ) ( ) ( )112

10 102 2 22

111 1 1 .2 22

xI x x dx x d x C

+= + = + + = +

c) S dng cc cng thc vi phn ( )

( )

32 31

3 3

2

xx dx d d x a

du d uu

= =

=

Ta c ( ) ( )3 32 3

3 3 3 3

1 11 2 2 1.

3 3 31 1 2 1

d x d xx dx xI Cx x x

+ + += = = = +

+ + +


a) 24 1I x x dx= b) 5 2 1dxIx

=

c) 6 5 2I x dx=

Ti liu tham kho:

02. PP VI PHN TM NGUYN HM Thy ng Vit Hng

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Hng dn gii:

a) S dng cc cng thc vi phn ( ) ( )2 2 2

1

1 12 2 2

1

nn

xxdx d d x d a x

uu du d

n

+

= = =

= +

Ta c ( ) ( ) ( ) ( ) ( )321 1

2 2 2 2 22 24

11 11 1 1 1 .2 2 3

xI x x dx x d x x d x C

= = = = +

b) S dng cc cng thc vi phn ( ) ( )( )

1 1ax ax

2

dx d b d ba a

du d uu

= + =

=

Ta c ( ) ( ) ( )25 52 1 2 11 2 1 .22 1 2 1 2 2 1du d u

ud x d xdxI I x C

x x x

=

= = = = +

c) S dng cc cng thc vi phn ( ) ( )

1

1 1ax ax

1

nn

dx d b d ba a

uu du d

n

+

= + =

= +

( ) ( ) ( ) ( ) ( )3 3

1 226

5 22 5 21 1 15 2 5 2 2 5 2 5 2 . .2 2 2 3 3

xxI x dx x d x x d x C C

= = = = + = +


a) 3

7 5 4

25

xI dxx

=

b) 8 5(3 2 )dxI

x=

c) 3

9ln xI dx

x=

Hng dn gii:

a) S dng cc cng thc vi phn ( ) ( )43 4 4

1

1 14 4 4

1

n

n

xx dx d d x a d a x

du udnu

+

= = =

=

+

( ) ( ) ( ) ( )4

4 444 55134 45

7 5 54 4

5 55 542 1 12 5 5 . .2 2 4 85 5

xdxxxI dx x d x C C

x x

= = = = + = +

b) Ta c ( ) ( ) ( )6

58 5

3 21 3 2 3 2 .(3 2 ) 2 12xdxI x d x C

x

= = = +

c) S dng cng thc vi phn ( )lndx d xx

= ta c ( )3 4

39

ln lnln ln .4

x xI dx x d x Cx

= = = +


a) ( )10 20103

4 2dxIx

=

b) 11cos xI dx

x= c) 12 cos sinI x x dx=

Hng dn gii:

a) Ta c ( ) ( ) ( )( )

( )2009

201010 2010 2009

4 23 3 3 34 2 4 2 .2 2 20094 2 4018 4 2

xdxI x d x C Cx x

= = = + = +

b) S dng cc cng thc vi phn ( )

( )cos sin

2

u du d udx d x

x

=

=

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Ta c ( )11 cos cos2 2 os 2sin .2x xI dx dx c x d x x Cx x= = = = + c) S dng cc cng thc vi phn ( )( )

cos sinsin x cos

u du d udx d x

=

=

Ta c ( ) ( ) ( )3

31 2212

2 cos 2 coscos sin cos cos .

3 3x xI x x dx x d x C= = = = +


a) 313 sin cosI x x dx= b) 14 5sin

cos

xI dxx

= c) 415 sin cosI x xdx= Hng dn gii:

a) S dng cc cng thc vi phn ( )( )sin coscos sin

u du d ux dx d x

=

=

Ta c ( ) ( ) ( )1 43 3

433 41 343 33 13

3 sinx 3 sinsin cos sinx sin

4 4

u du d uxI x x dx d x I C C

=

= = = + = +

b) Ta c ( )4

14 5 5 4

cossin (cos ) 1.

cos cos 4 4cosxx d xI dx C C

x x x

= = = + = +

c) S dng cc cng thc vi phn ( )

1

cos sin

1

nn

x dx d x

uu du d

n

+

=

= +

Khi ta c ( )5

4 554 4

15 15sin

sin cos sin sin .5

uu du d xI x x dx x d x I C

=

= = = +


a) 16 tanxI dx= b) 17 sin 4 cos 4I x x dx= c) 18sin

1 3cosx dxI

x=

+

Hng dn gii:

a) S dng cc cng thc sin x (cos )

ln

dx d xdu

u Cu

=

= +

Ta c ( )16 cossintan ln cos .cos cos

d xxdxI x dx x Cx x

= = = = +

b) Ta c ( ) ( )17 1 1sin 4 cos 4 sin 4 cos4 4 sin 4 sin 44 4I x x dx x x d x x d x= = = ( )3 322 sin 41 sin 4

. .

4 3 6x xC C= + = +

c) Ta c ( ) ( )18 cos 3cos 1sin 1 1 ln 1 3cos .1 3cos 1 3cos 3 1 3cos 3d x d xx dxI x C

x x x

+= = = = + +

+ + +


a) ( )19 22cos

2 5sinx dxI

x=

b) 20cos

4sin x 3x dxI =

c) ( )21 tan .ln cosI x x dx= Hng dn gii:

a) S dng cng thc vi phn 2

cos (sin x)1

xdx ddu d

uu

=

=

( )( )

( )( )

( ) ( )19 2 2 22 sin 2 5sin2cos 2 2

.

5 5 2 5sin2 5sin 2 5sin 2 5sind x d xx dxI C

xx x x

= = = = +

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b) S dng cng thc vi phn ( )cos (sin x)

2

xdx ddu d u

u

=

=

Ta c ( ) ( ) ( )20 sin 4sin 4sin 3cos 1 1 1 4sin x 3 .4 2 24sin x 3 4sin x 3 4sin x 3 2 4sin x 3d x d x d xxdxI C

= = = = = +

c) S dng cc cng thc nguyn hm c bn ( )

2

cossintan ln cos

cos cos

2

d xxdxxdx x C

x x

uu du C

= = = +

= +

Ta c ( ) ( ) ( ) ( ) ( ) ( )21 cossintan .ln cos ln cos ln cos ln cos ln coscos cos

d xxI x x dx x dx x x d xx x

= = = = = 2 2

21ln (cos ) ln (cos )

.

2 2x xC I C= + = +


a) 22 2tan

cos

xI dxx

= b) 3

23 4tancos

xI dxx

= c) 24 2tan 2 1cos 2

xI dxx

+=

Hng dn gii:

a) S dng cc cng thc ( )2

2

tancos

2

dx d xx

uu du C

=

= +

Ta c ( )2 2

22 222 2tan tan tan

tan . tan tan .2 2cos cos

x dx x xI dx x x d x C I Cx x

= = = = + = +

b) S dng cc cng thc ( )2

22

tancos

1 1 tancos

dx d xx

xx

=

= +

Ta c ( ) ( )3 3 3 2 5 323 4 2 2tan 1tan . . tan . 1 tan (tan ) tan tan (tan )cos cos cosx dxI dx x x x d x x x d xx x x= = = + = + 6 4 6 4

23tan tan tan tan

.

6 4 6 4x x x xC I C= + + = + +

c) S dng cc cng thc ( )2 2

2

1 ( ) 1tan( )

cos cos

2

dx d ax d axax a ax a

uu du C

= =

= +

Ta c 24 2 2 2 2 2tan 2 1 tan 2 1 tan 2 (2 ) 1 (2 )

2 2cos 2 cos 2 cos 2 cos 2 cos 2x xdx dx x d x d xI dx

x x x x x

+= = + = +

2 2

241 1 tan 2 tan 2 tan 2 tan 2

tan 2 (tan 2 ) (tan 2 ) .2 2 4 2 4 2

x x x xx d x d x C I C= + = + + = + +


a) 25 2cotsin

xI dxx

= b) 26 3tan

cos

xI dxx

= c) 27cot

picos

2

xI dxx

=

+

Hng dn gii:

a) S dng cc cng thc ( )2

2

cotsin

2

dx d xx

uu du C

=

= +

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Ta c ( )2 2

25 252 2cot cot cot

cot . cot cot .2 2sin sin

x dx x xI dx x x d x C I Cx x

= = = = + = +

b) S dng cc cng thc ( )

1

sin x cos

1

n

n

dx d xdu u Cu n

+

=

= + +

Ta c ( ) ( )3

26 263 4 4 3 3

cos costan sin 1 1.

cos cos cos 3 3cos 3cosd x xx xdxI dx C C I C

x x x x x

= = = = + = + = +

c) S dng cc cng thc

( )

2

cos sinpi

cos sin21

x dx d x

x x

du Cu u

=

+ =

= +

Ta c ( )27 272 2cot cos cos (sin ) 1 1

.

pi sin . sin sin sin sin sincos

2

x x x dx d xI dx dx C I Cx x x x x x

x

= = = = = + = +

+


a) 283 xeI dx

x= b)

tan 2

29 2cos

xe dxIx

+

= c) 21

30 .xI x e dx=

d) cos31 sinxI e x dx= e) 2ln 3

32

xeI dxx

+

=

Hng dn gii:

a) S dng cc cng thc ( )2u u

dx d xx

e du e C

=

= +

Ta c ( )28 283 3.2 6 6 6 .2x

x x x xe dxI dx e e d x e C I e Cx x

= = = = + = +

b) S dng cc cng thc ( ) ( )2 tan tancos

u u

dx d x d x kx

e du e C

= =

= +

Ta c ( )tan 2

tan 2 tan 2 tan 2 tan 229 292 2 tan 2 .cos cos

xx x x xe dx dxI e e d x e C I e C

x x

++ + + +

= = = + = + = +

c) S dng cc cng thc ( ) ( )2 21 1 1

2 2u u

x dx d x d x

e du e C

= =

= +

Ta c ( )2 2 2 2 21 1 1 2 1 130 301 1 1. 1 .2 2 2x x x x xI x e dx e x dx e d x e C I e C = = = = + = + d) S dng cc cng thc

( )sin cosu u

x dx d x

e du e C

=

= +

Ta c ( )cos cos cos cos31 31sin cos .x x x xI e x dx e d x e C I e C= = = + = +

e) S dng cc cng thc ( ) ( )ln ln

u u

dx d x d x kx

e du e C

= =

= +

Ta c ( ) ( )2ln 3

2ln 3 2ln 3 2ln 3 2ln 332

1 1ln 2ln 3 .2 2

xx x x xe dxI dx e e d x e d x e C

x x

++ + + +

= = = = + = +

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Vy 2ln 3

2ln 332

1.

2

xxeI dx e C

x

++

= = +

BI TP LUYN TP: 1) 1 21

xI dxx

=

+ 2) 2 10

2 (1 )I x x dx= + 3) 3cos xI dx

x=

4) 4 cos sinI x xdx= 5) 5 3sin

cos

xI dxx

= 6) 36 sin cosI x xdx=

7) 7 2 5xI dx

x=

+ 4) 8 2 1

dxIx

=

3) 9 5 2I xdx=

10) 3

10ln xI dx

x= 11)

2 111 .

xI x e dx+= 12) 412 sin cosI x xdx=

13) 13 5sin

cos

xI dxx

= 14) 14 cotI x dx= 15) 15 2tancos

xI dxx

=

16) tan

16 2cos

xeI dxx

= 17) 17xeI dxx

= 18) 218 1I x x dx= +

19) 19 5(3 2 )dxI

x=

20) 2 320 5I x x dx= + 21) 2

21 3 1x dxIx

=

+

22) 222 1I x x dx= 23) 23 cos 1 4sinI x x dx= + 24) 224 1I x x dx= +

25) cos25 sinxI e x dx= 26) 2 2

26 .xI x e dx+= 27) 27

sin1 3cos

x dxIx

=

+

28) 2128 . xI x e dx= 29) ( )sinx29 cos cosI e x x dx= + 30) 2ln 1

30

xeI dxx

+

=

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1. Vi phn nhm hm a thc, hm cn

3 4

1 (4 5 )I x x dx= = ....................................................................................................................................

32 3

2 2 1 3 )I x x dx= + = .................................................................................................................................

3 24 3 2xdxI

x= =

...........................................................................................................................................

5

4 61 5xI dx

x= =

..........................................................................................................................................

3

5 4

32 3

xI dxx

= =

+ ......................................................................................................................................

( )6 222 3xdxI

x= =

.........................................................................................................................................

2

7 cos(3 4 )I x x dx= = ................................................................................................................................

3 4

8 sin(1 5 )I x x dx= + = ...............................................................................................................................

24 5

9xI xe dx += = ..........................................................................................................................................

4

10 2

xe dxIx

= = ................................................................................................................................................

3

11 2

xe dxIx

= = ..............................................................................................................................................

12 3dxI

x x= =

+ ...........................................................................................................................................

2. Vi phn nhm hm lng gic

3

1 sin .cosI x xdx= = ...................................................................................................................................

5

2 cos .sinI x xdx= = ...................................................................................................................................

3 sin . 3cos 2I x x dx= + = .........................................................................................................................

4

4 cos . 5 2sinI x xdx= = ..........................................................................................................................

5sin

2 5cosxdxI

x= =

+......................................................................................................................................

Ti liu bi ging:

02. PP VI PHN TM NGUYN HM Thy ng Vit Hng

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6sin

1 3cosxdxI

x= =

......................................................................................................................................

( )7 2cos

1 2sinxdxI

x= =

.....................................................................................................................................

8sin 2

7 2cos 2xdxI

x= =

......................................................................................................................................

9sin 3

1 2cos3xdxI

x= =

+ .....................................................................................................................................

10 2tan3cos

xdxIx

= = ...........................................................................................................................................

11 4tancos

xdxIx

= = ............................................................................................................................................

3cos 2

12 sin .xI x e dx= = .................................................................................................................................

2 5sin 2

13 cos 2 .xI x e dx= = .............................................................................................................................

2cot 1

14 2sin

xeI dxx

= = ........................................................................................................................................

15 2sin 4cot 3dxI

x x= =

...........................................................................................................................

3. Vi phn nhm hm m, loga

1 2 1

x

x

eI dxe

= =

.........................................................................................................................................

3

2 31 5

x

x

eI dxe

= =

.....................................................................................................................................

( )2

3 221 3

x

x

eI dxe

= =

..................................................................................................................................

3

4ln xI dx

x= = ...........................................................................................................................................

5 1 5lndxI

x x= =

.....................................................................................................................................

( )6 22 3lndxI

x x= =

+ ..................................................................................................................................

7 2

ln1 4ln

xdxIx x

= =

...................................................................................................................................

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Dng 1. i bin s cho cc hm v t

Phng php gii:

Nu hm f(x) c cha ( )n g x th t 1( ) ( ) . '( )n nnt g x t g x n t g x dx= = =

Khi , ( ) ( )I f x dx h t dt= = , vic tnh nguyn hm ( )h t dt n gin hn so vi vic tnh ( ) .f x dx MT S V D MU: V d 1. Tm nguyn hm ca cc hm s sau:

a) 1 4 1xdxIx

=

+ b) 3 22 2I x x dx= + c)

2

3 1x dxI

x=

Hng dn gii:

a) t

2

2 221

12 4. 14 24 1 4 1 ( 1)1 84 1

4

t tdttdt dxxdx

t x t x I t dtt txx

=

= + = + = = =

+=

33 (4 1)1 1 4 1 .8 3 8 3

xtt C x C

+ = + = + +

b) t 2 2 2 2 2 3 2 22 2 2 2 2 . ( 2).t x t x x t xdx tdt x dx x xdx t tdt= + = + = = = =

Khi ( ) ( ) ( ) ( )5 32 25 3

2 3 2 4 22

2 2 22 . . 2 2 2.

5 3 5 3

x xt tI x x dx t t tdt t t dt C C+ +

= + = = = + = +

c) t ( )( )2222 2

2 32 2

2 1 .1 1 1 2

1 1

dx tdt t tdtx dxt x t x x t I

tx t x

=

= = = = = =

( ) ( ) 5 35 322 4 2 (1 ) 2 (1 )22 1 2 2 1 2 2 15 3 5 3x xt t

t dt t t dt t C x C = = + = + + = + +

Khi ( ) ( ) ( ) ( )5 32 25 3

2 3 2 4 22

2 2 22 . . 2 2 2. .

5 3 5 3

x xt tI x x dx t t tdt t t dt C C+ +

= + = = = + = +


a) 4ln1 ln

x dxIx x

=

+ b)

2

5 3ln

2 lnxdxI

x x=

c) 6ln 3 2lnx x dxI

x

+=

Hng dn gii:

a) t ( )2 2

24

ln 1 1 .2ln1 ln 1 ln1 ln2

x t t tdtx dxt x t x Idx

x txtdtx

=

= + = + = =+=

( ) 3 332 4(1 ln ) 2 (1 ln )2 1 2 2 1 ln 2 1 ln .3 3 3x xt

t dt t C x C I x C + + = = + = + + = + +

b) t 3

2 3 2 233

52 3

ln 2 ln (2 ) .32 ln 2 ln .2 ln3

x tx dx t t dt

t x t x Idxx txt dt

x

=

= = = ==

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03. PP I BIN S TM NGUYN HM P1 Thy ng Vit Hng

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( ) 8 58 5 3 37 4 2 23(2 ln ) 4 (2 ln )43 4 4 3 2 3 2 (2 ln )8 5 8 5x xt t

t t t dt t C x C

= + = + + = + +

c) t

2

2

3ln23 2ln 3 2ln

2 2

tx

t x t xdx

tdtx

== + = +

=

T ta c ( )2 4 26 ln 3 2ln 3 1ln 3 2ln . . . 32 2x x dx dx tI x x t tdt t t dtx x +

= = + = =

( ) ( ) ( ) ( )5 3 5 35 5 336

3 2ln 3 2ln 3 2ln 3 2ln1.

2 5 10 2 10 2 10 2x x x xt t t

t C C C I C+ + + +

= + = + = + = +


a) 71x

dxIe

=

b) ( )2

8 31

x

x

e dxIe

=

+ c) 9 2 4

dxIx x

=

+ d) 10 4 1

dxIx x

=

+

Hng dn gii:

a) t 2

22

2

111 1 2

21

xx

x x

x

e te t

t e t e tdtdxe dx tdtt

= = = =

==

Khi 7 2 22 2 2 ( 1) ( 1)

( 1)( 1) ( 1)( 1) 1 1.( 1) 11xdx tdt dt dt t t dt dtI dt

t t t t t tt t te

+ = = = = = =

+ + +

71 1 1 1 1ln 1 ln 1 ln ln ln .1 1 1 1 1

x x

x x

t e et t C C C I C

t e e

= + + = + = + = ++

+ +

b) t ( ) ( )

( )22 228 33 3

1 .21 .1 12 1 1

x x x xx x

xx x

t tdte t e dx e e dxt e t e I

te dx tdt e e

= = + = + = = =

= + +

( )2 23 2 2

1 .2 1 1 12 2 2 2 1 .1

x

x

t tdt t dtdt dt t C e Ctt t t e

= = = = + + = + + + +

c) t 2 2

2 22 2 2

2 2

444 42 2

4

x tx t

t x t x dx xdx tdtxdx tdt

x x t

= = = + = +

= = =

Khi , 9 2 22 21 1 1 ( 2) ( 2) 1

.

4 ( 2)( 2) 4 2 24 44 4dx dx tdt dt t t dt dtI dt

x t t t t tt tx x x

+ = = = = = = + + + +

( ) 2 292 21 1 2 1 4 2 1 4 2ln 2 ln 2 ln ln ln .4 4 2 4 44 2 4 2t x x

t t C C C I Ct x x

+ + = + + = + = + = +

+ + + + +

d) t 4 2

4 24 2 4 3

34 2

11

1 14 2

2( 1)

x tx t

t x t x dx x dx tdtx dx tdt

x x t

= =

= + = + = ==

Khi , 10 2 24 41 1 1 1 ( 1) ( 1)

. .

2 4 ( 1)( 1)2( 1) 11 1dx dx tdt dt t tI dt

x t t tt tx x x

+ = = = = =

+ + +

( ) 44

1 1 1 1 1 1 1ln 1 ln 1 ln ln .4 1 1 4 4 1 4 1 1

dt dt t xt t C C C

t t t x

+ = = + + = + = +

+ + + +


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a) 11 1 2 5dxI

x=

+ b) 12 21 2

x dxIx

=

+

c) 3

13 3 24x dxI

x=

+ d)

2

141 4ln lnx xI dx

x

+=

Hng dn gii:

a) t 2 22 5 2 5 2 55tdt

t x t x tdt dx dx= = = =

Khi , ( )11 2 2 1 1 2 1 21 ln 15 1 5 1 5 1 51 2 5dx t dt tI dt dt t t C

t t tx

+ = = = = = + + + + ++

( )11 2 2 5 ln 2 5 1 .5I x x C = + + b) t 2 2 22 2 2 2t x t x tdt xdx xdx tdt= + = + = = Khi , 12 2

1 (1 ) 1 (1 )1 ln 11 1 1 11 2

x dx t dt t d tI dt dt dt t t Ct t t tx

= = = = = = +

+

2 212 ln 1 2 2 .I x x C = + + +

c) t ( )2 3

2 33 2 3 2 3 3 22

2

44 34 4 43 23 22

x tx t

t x t x x dx t t dtt dtt dt xdx xdx

= = = + = + =

= =

( ) ( ) ( ) ( )5 22 23 2 3 33 5

4 213 3 2

3 4 3 443 3 34 2 .2 2 2 5 10 44

x xt t dtx dx tI t t dt t C Ctx

+ +

= = = = + = + +

d) t 2 2 2 ln1 4ln 1 4ln 2 4.2ln .4

dx x dx tdtt x t x tdt x

x x= + = + = =

( )3232 214

1 4lnln 11 4ln . .4 4 12 12

xx dx tdt tI x t t dt C Cx

+ = + = = = + = +

BI TP T LUYN:

1) 14 3

1xI dx

x

=

+ 2) 2 2 1

xdxIx

=

+

3) 31xI dx

x

+= 4) 4 1 1 3

dxIx

=

+ +

5) 7 1 2 1xdxI

x=

+ 6) 3 26 1I x x dx=

7) 37 4I x x dx= + 8) 28 3 2I x x dx=

9) 3

9 3 21x dxI

x=

+ 10) 10 3 1

dxIx x

=

+

11) 11 3 2 4dxI

x x=

+ 12) 12

1 3ln lnx xI dxx

+=

13) 2

131 1

x

x

e dxIe

=

+ 14) ( )14 21

dxIx x

=

+

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Dng 2. PP lng gic ha

Nu hm f(x) c cha 2 2a x th t 2 2 2 2 2

(a sin ) cosa sin

sin cos

dx d t a t dtx t

a x a a t a t

= ==

= =

Nu hm f(x) c cha 2 2a x+ th t 2

2 2 2 2 2

( tan )cos

tan

tancos

= =

= + = + =

adtdx d a tt

x a ta

a x a a tt

MT S V D MU: V d 1. Tm nguyn hm ca cc hm s sau:

a) ( )1 2 ; 24= =dxI a

x b) ( )22 1 ; 1= =I x dx a

c) ( )23 2 ; 11= =x dxI a

x d) ( )2 24 9 ; 3= =I x x dx a

Hng dn gii:

a) t 12 2 2(2sin ) 2cos 2cos2sin

2cos4 4 4sin 2cos 4

dx d t t dt dx t dtx t I dt t C

tx t t x

= == = = = = +

= =

T php t 12sin arcsin arcsin2 2x x

x t t I C = = = +

b) t 2 2

(sin ) cossin

1 1 sin cos

dx d t t dtx t

x t t

= ==

= =

Khi 221 cos2 1 1 11 cos .cos cos2 sin 2

2 2 2 2 4t tI x dx t t dt dt dt t dt t C+= = = = + = + +

T 2 2

2cos 1 sin 1sin sin 2 2sin .cos 2 1arcsint t x

x t t t t x xt x

= = = = =

=

22

arcsin 1 12 2

xI x x C = + +

c) t 2 2

(sin ) cossin

1 1 sin cos

dx d t t dtx t

x t t

= ==

= =

Khi , 2 2

23 2

sin .cos 1 os2 1 1sin sin 2

cos 2 2 41x dx t t dt c tI t dt dt t t C

tx

= = = = = +

T 2 2

2cos 1 sin 1sin sin 2 2sin .cos 2 1arcsint t x

x t t t t x xt x

= = = = =

=

23

arcsin 1 12 2

xI x x C = +

d) t 2 2

(3sin ) 3cos3sin

9 9 9sin 3cos

dx d t t dtx t

x t t

= ==

= =

Khi , 2 2 2 2 2 2481 81 1 os49 9sin .3cos .3cos 81 sin .cos sin 24 4 2

c tI x x dx t t t dt t t dt t dt dt= = = = =

Ti liu bi ging:

03. PP I BIN S TM NGUYN HM P2 Thy ng Vit Hng

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81 1 1 81 1os4 sin 4

4 2 2 4 2 8tdt c t dt t C = = +

T

22

2cos 1 sin 1 293sin sin 2 13 9

arcsin3

xt t

x xx t t

xt

= =

= =

=

Mt khc, 2 2 2 2

2 2 2 2os2 1 2sin 1 2 1 sin 4 2sin 2 . os2 2. 1 . 13 9 3 9 9x x x x x

c t t t t c t

= = = = =

T ta c 2 2

4

arcsin81 23 1 . 1 .4 2 6 9 9

x

x x xI C

= +


a) ( )1 2 ; 11dxI a

x= =

+ b) 2

2 2 5I x x dx= + + c) ( )2

3 2; 2

4x dxI ax

= =

+

Hng dn gii:

a) t 2 2

21 2

2 2

(tan ) (1 tan ) (1 tan )tan cos

1 tan1 1 tan

dtdx d t t dt t dtx t I dt t Ct

tx t

= = = + +

= = = = ++ + = +

T gi thit t 1tan arctan arctan .x t t x I x C= = = +

b) Ta c 12 2 22 2 5 ( 1) 4 ( 1) 4t xI x x dx x d x I t dt= += + + = + + + = +

t 2

2 222 2

2(2 tan ) 2 coscos2 tan 22 cos cos.cos4 4 4 tan

coscos

dudt d u du du u duut u Iu u

ut uuu

= =

= = = = + = + =

2(sin ) 1 (1 sin ) (1 sin ) 1 (sin ) 1 (sin ) 1 1 sin(sin ) ln .

1 sin 2 (1 sin )(1 sin ) 2 1 sin 2 1 sin 2 1 sind u u u d u d u ud u C

u u u u u u

+ + += = = + = +

+ +

T php t 2 2

2 22 2 2

1 42 tan tan 1 sin 1 os 12 os 4 4 4t t t

t u u u c uc u t t

= = = + = = =+ +

T ta c 2 2

2

2 2

11 11 1 sin 1 14 2 5ln ln ln .12 1 sin 2 21 1

4 2 5

t x

u t x xI C C Ct xu

t x x

++ +

+ + + += + = + = +

+

+ + +

c) t 2

2

2 2

2(2 tan ) 2(1 tan )os2 tan

4 4 tan 4

dtdx d t t dtc tx t

x t

= = = +

= + = +

( )2 2 2 2 2

2 23 23 42 2

4 tan .2(1 tan ) sin sin .cos sin . (sin )4 tan 1 tan 4 4 4cos cos2 1 tan 1 sin

t t dt t t t dt t d tI t t dt dtt tt t

+ = = + = = =

+

t ( )222

3 2 22

1 (1 ) (1 )sin 4 4 4

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

u t I du du duu u uu

+ = = = =

+

2

2 2 2 21 1 2 (1 ) (1 ) (1 ) (1 )

1 1 (1 ) (1 ) (1 )(1 ) (1 ) (1 ) (1 )(1 )du du du d u d u u u dudu

u u u u u u u u u u

+ + + = = + = +

+ + + + +

1 1 1 1 1 1 1 1 ln 1 ln 11 1 1 1 1 1 1 1 1 1

du dudu u u Cu u u u u u u u u u

+ = = + + +

+ + + + +

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31 1 1 1 1 1 1 1 sin 1ln ln ln .

1 1 1 1 1 1 sin 1 sin 1 sin 1u u tC I C C

u u u u u u t t t

= + + = + + = + + + + + + + +

T gi thit 2 2

2 2 22 2 2

1 42 tan tan 1 tan 1 os sin2 os 4 4 4x x x

x t t t c t tc t x x

= = = + = + = =+ +

2

32

2 2 2

11 1 4

sin ln .4 1 1 1

4 4 4

x

x xt I Cx x xx

x x x

+ = = + ++

+ ++ + +


a) 1 2 1dxI

x=

b) 2 2 2 4dxI

x x=

c) 3 2 2 2dxI

x x=

Hng dn gii:

a) t 2

21 22

222

1 coscos

sin sin1 cossin

sin sin .cot1 11 cot1 1sin

t dtt dtdx d dxt t dx t dttx I

t t txx tx

t

= =

=

= = =

= =

2 2sin (cos ) (cos ) 1 (1 cos ) (1 cos ) 1 1 cos(cos ) ln .sin 1 cos (1 cos )(1 cos ) 2 (1 cos )(1 cos ) 2 1 cos

t dt d t d t t t td t Ct t t t t t t

+ + += = = = = +

+ +

T php t

2

22 2

12 2

111 1 1 1os 1 sin 1 cos ln .

sin 2 11

x

x xx c t t t I Ct x x x

x

+

= = = = = +

b) t 2 2

2 2 2222

2 2cos 2cossin sin2 sin

8cotsin 4 4 2cot 44 4sinsin

t dt t dtdx d dxt t t

xtt

x t x xxtt

= = = =

= =

=

Khi , 2 2 2 22

2cos 1 1sin cos .8cot 4 44 sin .

sin

dx t dtI t dt t Ctx x tt

= = = = +

T 2 2

2 222

2 4 4 4os 1 sin 1 cos .

sin 4x x

x c t t t I Ct x x x

= = = = = +

c) ( )

13 32 2 2 22

( 1)2 2 ( 1) 3 3 3

t xdx d x dt dtI Ix x x t t

=

= = = =

t 2

2

222

3 3 cos3 cos

sin sin3sin

sin 3 3 3 cot3 3sin

u dudt d u dudtu ut u

ut ut

u

= = = =

=

=

3 2 222

3 cos sin (cos ) (cos )sin 1 cos (1 cos )(1 cos )sin . 3 cot3

dt u du u du d u d uIu u u uu ut

= = = = = +

1 (1 cos ) (1 cos ) 1 1 cos(cos ) ln .2 (1 cos )(1 cos ) 2 1 cos

u u ud u Cu u u

+ + += = +

+

T

2 2

22

32 2 2

3 2 21 13 3 3 1 1 1os 1 cos ln ln .sin 2 23 2 21 1

1

t x xt t xt c u t I C C

u t t t x xt x

+ +

= = = = + = +

Ch : Tng hp cc kt qu ta thu mt s kt qu quan trng sau:

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2 21

arc tan .dx x C

x a a a

= + +

2 21 ln .

2dx x a C

x a a x a

+= +

2 21 ln .

2dx x a C

a x a x a

= + +

2

2ln .dx x x a C

x a= + +

BI TP LUYN TP:

1) 2

1 2 4x dxIx

=

+ 2)

2

2 21 xI dxx

= 3) 2

3 24x dxI

x=

4) 4 21

3 2I dx

x x=

5) 25 2 1I x dx= + 6) 6 22 5dxIx

=

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Xt nguyn hm ca hm phn thc hu t ( )( )P xI dxQ x=

Nguyn tc gii: Khi bc ca t s P(x) ln hn Q(x) th ta phi chia a thc quy v nguyn hm c bc ca t s nh hn mu s. I. MU S L BC NHT

Khi Q(x) = ax + b. Nu bc ca P(x) ln hn th ta chia a thc. Khi P(x) l hng s (bc bng 0) th ta c ( ) (ax ) ln ax .( ) ax ax

P x k k d b kI dx dx b CQ x b a b a+

= = = = + ++ +


a) 14

2 1I dx

x=

b) 211

xI dxx

+=

c) 32 13 4

xI dxx

+=

d) 2

44

3x xI

x

+ +=

+

Hng dn gii:

a) Ta c 14 4 (2 1) 2ln 2 1 .

2 1 2 2 1d xI dx x C

x x

= = = +

b) 21 1 2 21 2 2ln 1 .1 1 1 1

x x dxI dx dx dx dx x x Cx x x x

+ + = = = + = + = + +

c) ( )

( )( )

3

1 53 4 3 42 1 1 5 1 5 1 52 23 4 3 4 2 2 3 4 2 2 3 4 2 8 3 4

x d xx dxI dx dx dx x xx x x x x

+ +

= = = + = + =

31 5 1 5ln 3 4 ln 3 4 .2 8 2 8

x x C I x x C= + = +

d) ( ) ( )2 2

4

34 102 2 10 2 10ln 3 .3 3 3 2

d xx x xI x dx x dx x x Cx x x

++ + = = + = + = + + + + + +


a) 3

57

2 5x xI dx

x

+=

+ b)

3 2

63 3 2

1x x xI dx

x

+ + +=

c) 4 2

74 3 2

2 1x x xI dx

x

+ + +=

+

Hng dn gii:

a) Chia t s cho mu s ta c 3

2

497 1 5 21 8

2 5 2 4 8 2 5x x

x xx x

+= +

+ +

Khi 3

2 25

497 1 5 21 1 5 21 498

2 5 2 4 8 2 5 2 4 8 8 2 5x x dxI dx x x dx x x dx

x x x

+ = = + = + + + +

( )3 2 3 22 51 5 21 49 5 21 49. . ln 2 5 .

2 3 4 2 8 16 2 5 6 8 8 16d xx x x x x

x x Cx

+= + = + + +

+

b) Ta c 3 2

2 3 26

3 3 2 93 6 7 3 7 9ln 1 .1 1

x x xI dx x x dx x x x x Cx x

+ + + = = + + + = + + + +

c) Chia t s cho mu s ta c 4 2

3 2

54 3 2 1 22 2

2 1 2 2 1x x x

x x xx x

+ + += + +

+ +

Ti liu bi ging:

04. NGUYN HM CA HM HU T - P1 Thy ng Vit Hng

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Khi 4 2

3 2 3 27

54 3 2 1 1 522 2 2 2

2 1 2 2 1 2 2 2 1x x x dxI dx x x x dx x x x dx

x x x

+ + +

= = + + = + + + + +

( )4 3 4 32 22 11 5 1 52. ln 2 1 .4 3 2 4 2 1 2 3 2 4

d xx x x xx x x x x C

x

+= + + = + + + +

+

BI TP LUYN TP:

1) 12 1

3xI dx

x

=

+ 2)

2

23 1

1x xI dx

x

+ =

+ 3)

3 2

33 3 2

1x x xI dx

x

+ + +=

4) 3

47

2 5x xI dx

x

+=

+ 5) 5

14 3xI dx

x

+=

4) 4 2

65 3

3 1x x xI dx

x

+=

+

II. MU S L TAM THC BC HAI

Khi Q(x) = ax2 + bx + c. Ta c ba kh nng xy ra vi Q(x). TH1: Q(x) = 0 c 2 nghim phn bit x1 v x2 Nu P(x) l hng s th ta s dng thut phn tch t s c cha nghim ca mu s. Nu P(x) bc nht th ta c phn tch ( )( ) ( )( )1 2 1 2 1 2

( ) ( ) 1( ) ( )P x P x A BQ x a x x x x Q x a x x x x a x x x x

= = = +

ng nht h s hai v ta c A, B. T , quy v bi ton nguyn hm c mu s l hm bc nht xt trn. Nu P(x) c bc ln hn hoc bng 2 th ta chia a thc, quy bi ton v hai trng hp c bc ca P(x) nh trn gii. Ch : Vic phn tch a thc thnh nhn t vi cc phng trnh bc hai c h s a khc 1 phi theo quy tc

( )( )2 1 2+ + = ax bx c a x x x x

V d: 2( 1)(3 1) : '.

3 4 1 1( 1) : .3

x x dungx x

x x sai

+ =

Khi t s l bc nht th ngoi cch ng nht trn, ta c th phn tch t s c cha o hm ca mu, ri tch thnh 2 nguyn hm (xem cc v d di y). V d 1. Tm nguyn hm ca cc hm s sau:

a) 1 2 2 3dxI dx

x x=

b) 2 22

3 4 1dxI

x x=

+

c) 3 22 3

3 4xI dx

x x

+=

d) 4 23 4

5 6 1xI dx

x x

+=

+ +

Hng dn gii:

a) 1 21 ( 1) ( 3) 1 1 3ln .( 1)( 3) 4 ( 1)( 3) 4 3 1 4 12 3

dx dx x x dx dx xI dx dx Cx x x x x x xx x

+ = = = = = + + + + +

b) 2 2 22 2 (3 1) 3( 1)2 2

3 4 1 3 4 1 ( 1)(3 1) 4 ( 1)(3 1)dx dx dx x xI dx

x x x x x x x x

= = = =

+ +

1 1 1 (3 1) 1 1 1 3 13 ln 1 ln 1 ln 3 1 ln .2 1 3 1 2 2 3 1 2 2 2 1

dx dx d x xx x x C C

x x x x

= = + = + + = +

c) 3 22 3

3 4xI dx

x x

+=

Cch 1:

Nhn thy mu s c hai nghim x = 1 v x = 4, khi ( )( )22 3 2 3

1 4 1 43 4x x A B

x x x xx x

+ += = +

+ +

ng nht ta c ( ) ( )1

2 52 3 4 13 4 11

5

AA Bx A x B x

A B B

= = +

+ + + = +

=

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T 3 2

1 112 3 1 11 1 115 5 ln 1 ln 4 .

1 4 5 1 5 4 5 53 4x dx dxI dx dx x x C

x x x xx x

+

= = + = + = + + + + +

Vy 31 11ln 1 ln 4 .5 5

I x x C= + + +

Cch 2: Do mu s c o hm l 2x 3 nn ta s phn tch t s c cha o hm ca mu nh sau:

( )23 2 2 2 2 2

3 42 3 2 3 6 (2 3) 6 6 ( 1)( 4)3 4 3 4 3 4 3 4 3 4d x xx x x dx dx dxI dx dx

x xx x x x x x x x x x

+ + = = = + = +

+

2 2 26 ( 1) ( 4) 6 6 4ln 3 4 ln 3 4 ln 3 4 ln .5 ( 1)( 4) 5 4 1 5 1

x x dx dx xx x dx x x x x C

x x x x x

+ = + = + = + + + + +

Nhn xt: Nhn hai cch gii, thot nhn chng ta lm tng l bi ton ra hai p s. Nhng, ch bng mt vi php bin i logarith n gin ta c ngay cng kt qu. Tht vy, theo cch 2 ta c:

2 6 4 6 6 1 11ln 3 4 ln ln 4 ln 1 ln 4 ln 1 ln 1 ln 4 .5 1 5 5 5 5

xx x x x x x C x x

x

+ = + + + + + = + + +

R rng, chng ta thy ngay u im ca cch 2 l khng phi ng nht, v cng khng cn dng n giy nhp ta c th gii quyt nhanh gn bi ton, v l iu m ti mong mun cc bn thc hin c!

d) 4 23 4 3 4

5 6 1 ( 1)(5 1)x xI dx dx

x x x x

+ += =

+ + + +

Cch 1:

Ta c

13 53 4 43 4 (5 1) ( 1)4 17( 1)(5 1) 1 5 1

4

AA Bx A Bx A x B x

A Bx x x x B

= = ++

= + + + + + = ++ + + +

=

T 43 4 1 17 1 17

( 1)(5 6) 4( 1) 4(5 1) 4 1 4 5 1x dx dxI dx dx

x x x x x x

+= = + = +

+ + + + + +

41 17ln 1 ln 5 1 .4 20

I x x C = + + + +

Cch 2: Do mu s c o hm l 10x + 6 nn ta s phn tch t s c cha o hm ca mu nh sau:

( ) ( )4 2 2 2 2

3 2210 6 10 63 4 3 2210 105 6 1 5 6 1 10 5 6 1 10 5 6 1

x xx dxI dx dx dxx x x x x x x x

+ + ++= = = +

+ + + + + + + +

( )2 22

5 6 13 22 3 22 (5 1) 5( 1)ln 5 6 110 5 6 1 10 (5 1)( 1) 10 40 (5 1)( 1)

d x x dx x xx x dx

x x x x x x

+ + + += + = + +

+ + + + + +

2 23 22 5 3 11 1ln 5 6 1 ln 5 6 1 ln .10 40 1 5 1 10 20 5 1

dx dx xx x x x C

x x x

+ = + + = + + + + + +


a) 3

5 24 2 1

1x xI dx

x

+ =

b) 6 25

3 2xI dx

x x

=

Hng dn gii:

a) Do t s c bc ln hn mu nn chia a thc ta c 3

5 2 24 2 1 6 14

1 1x x xI dx x dx

x x

+ = = +

Ta c 2

766 1 6 1 26 1 ( 1) ( 1)

1 51 ( 1)( 1) 1 12

AA Bx x A Bx A x B x

A Bx x x x x B

== +

= = + + + = + + +

=

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( ) ( )2

57 5 7 54 2 ln 1 ln 1 .

2 1 2 1 2 2I x dx x x x C

x x

= + + = + + + + +

b) Ta c 2 25 5 5 5 ( 3) ( 1)

3 2 2 3 ( 1)( 3) 1 3x x x A B

x A x B xx x x x x x x x

= = = + + + + + +

6 2

1 1 5 1 2 25 3 2 1 3 1 33 2

A B A x dx dxI dx dxA B B x x x xx x

= + = = = + = +

= = + +

( ) ( )2 26

3 3ln 1 2ln 3 ln ln .

1 1x x

x x C C I Cx x

= + + + = + = +

BI TP LUYN TP:

1) 1 22 1

3 2xI dx

x x

=

+ + 2) 2 2

3 45 6 1

xI dxx x

+=

+ + 3)

2

3 23 1

2 3 1xI dx

x x

+=

+ +

4) 4 25 4

3 2xI dx

x x

+=

5) 5 25 3

2 1xI dx

x x

+=

6) 6 21 5

4 5 1xI dx

x x

=

+ +

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Nguyn tc gii: Khi bc ca t s P(x) ln hn Q(x) th ta phi chia a thc quy v nguyn hm c bc ca t s nh hn mu s. II. MU S L TAM THC BC HAI (tip theo)

Khi Q(x) = ax2 + bx + c. Ta c ba kh nng xy ra vi Q(x). TH1: Q(x) = 0 c 2 nghim phn bit x1 v x2 TH2: Q(x) = 0 c nghim kp

Khi Q(x) c biu din di dng ( ) ( )2

2( )( ) = + =+

P xQ x ax b I dx

ax b

Nu P(x) l hng s th ta s dng cc bin i sau ( )

2

1

1

= +

= +

dx d ax ba

du Cuu

Nu ( )( )

( ) ( )2 2 2( )+ +

+ = + = = = + + + + +

m bmax b n

mx n m dx bm dxa aP x mx n I dx dx na ax b aax b ax b ax b

( ) ( )( )2 2 2 2

1ln .+ +

= + = + + + + +

bmnd ax b d ax bm m na bma ax b C

ax b a ax ba a aax b

Nu P(x) c bc ln hn hoc bng 2 th ta chia a thc, quy bi ton v hai trng hp c bc ca P(x) nh trn gii. Ch :

Ngoi cch gii nu trn, dng nguyn hm ny c cch gii tng qut l t t b

xt ax b a

dt adx

=

= + =


a) 1 22

2 1dxI

x x=

+ b) 2 26 9 1dxI

x x=

+ + c) 3 225 10 1

dxIx x

=

+

Hng dn gii:

a) 1 12 2 22 ( 1) 2 22 2 .

1 12 1 ( 1) ( 1)dx dx d xI C I C

x xx x x x

= = = = + = + +

b) 2 22 2 21 (3 1) 1 1

.

6 9 1 (3 1) 3 (3 1) 3(3 1) 3(3 1)dx dx d xI C I C

x x x x x x

+= = = = + = +

+ + + + + +

c) 3 32 2 21 (5 1) 1 1

.

25 10 1 (5 1) 5 (5 1) 5(5 1) 5(5 1)

= = = = + = + + dx dx d xI C I C

x x x x x x


a) 4 22 1

4 4 1xI dx

x x

=

+ + b) 2

5 24 3

4 12 9xI dx

x x

=

+ + c) 6 2

1 59 24 16

xI dxx x

=

+

Hng dn gii:

a) ( )4 2 22 1 2 1

4 4 1 2 1x xI dx dx

x x x

= =

+ + +

Ti liu bi ging:


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Cch 1:

t ( )4 2 2 22 1 2 1 2 1 2 1 12 1 ln

2 2 2 22 1

x t x t dt dt dtt x I dx t C

dt dx t tt tx=

= + = = = = + + = +

41 1ln 2 1 .2 2 1

I x Cx

= + + ++

Cch 2:

( ) ( )( )

( ) ( )( )

2

4 2 2 2 2 2 2

1 8 4 2 4 4 18 4 2 12 1 1 14 24 44 4 1 4 4 1 4 4 1 4 4 12 1 2 1

x d x xx d xx dxI dx dx dxx x x x x x x xx x

+ + ++ +

= = = =

+ + + + + + + ++ +

( ) ( )( )

22

2 2

4 4 1 2 11 1 1 1 1ln 4 4 1 ln 2 1 .4 4 2 1 2 2 14 4 1 2 1

d x x d xx x C x C

x xx x x

+ + += = + + + + = + + +

+ ++ + +

b) ( )( )

( )2

5 2 22 2

2 34 3 12 12 61 12 6 .4 12 9 4 12 9 2 32 3 2 3

d xx x dxI dx dx dx x x Cx x x x xx x

+ +

= = = = = + + + + + + + + +

c) ( )6 2 21 5 1 5

9 24 16 3 4x xI dx dx

x x x

= =

+

Cch 1:

t ( )6 2 2 25( 4)4 11 5 1 5 1733 4 3

3 93 43

ttx x dt t

t x I dx dtt txdt dx

++

= +

= = = = =

61 17 1 17 5 175ln 5ln 3 4 ln 3 4 .9 9 3 4 9 9(3 4)t C I x C x Ct x x

= + = + = + +

Cch 2:

( )( )( ) ( )

( ) ( )( )6 2 2 2 2

5 173 4 3 4 3 41 5 5 17 5 173 33 3 4 3 9 3 4 93 4 3 4 3 4 3 4

x d x d xx dx dxI dx dxx xx x x x

= = = =

( )65 17 1 5 17ln 3 4 . ln 3 4 .9 9 3 4 9 9 3 4

x C I x Cx x

= + + = + +

TH3: Q(x) = 0 v nghim

Khi , Q(x) c biu din di dng ( )2 2

22 24( )2 4

= + + = + + + +

b ac bQ x ax b c a x mx n ka a

Nu P(x) l hng s th ta s dng cc bin i sau ( )

2 2

1

1arctan

= +

= + +

dx d ax ba

du u Ca au a

Nu P(x) = x + th ta c phn tch sau: ( ) ( )

2 2 2 2

2 2 2 2 2 2

+ + ++ = = = +

+ + + + + + + +

bax b ax b dxx b dxa aI dx dx dx

a aax bx c ax bx c ax bx c ax bx c

( )2 22 2 22 2

2

2 ln

2 2 24 42 4 2 4

+ + = + = + + +

+ + + + + +

bd ax bx c b dx dxadx ax bx c

a a a aax bx c b ac b b ac ba x x

a a a a

2 22 2 2 2

2

2 22 22ln ln arctan .2 24 4 4

2 4

b bb d xax ba aaax bx c ax bx c C

a a ab ac b ac b ac bx

a a

+

+ = + + + = + + + +

+ +

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Nu P(x) c bc ln hn hoc bng 2 th ta chia a thc, quy bi ton v hai trng hp c bc ca P(x) nh trn gii. Nhn xt: Nhn vo biu thc ca bi ton tng qut trn c th ban u lm cho cc bn pht hong, nhng ng qu bn tm n n, bn ch cn nm c tng thc hin ca n l phn tch t s c cha o hm ca mu s, ri tch thnh hai bi ton nh hn u thuc dng n gin hc.


a) 1 2 2 3dxI

x x=

+ + b) 2 24 4 2dxI

x x=

+ + c) 3 29 24 20

dxIx x

=

+ +

Hng dn gii:

a) ( )( )

( ) ( )1 2 2 221 1 1

arctan .2 3 2 21 2 1 2

d xdx dx xI Cx x x x

+ + = = = = +

+ + + + + +

b) ( )( )

( ) ( )2 2 22 22 11 1

arctan 2 1 .4 4 2 2 22 1 1 2 1 1

d xdx dxI x Cx x x x

+= = = = + +

+ + + + + +

c) ( )( )

( )3 2 2 2 23 4 1 3 4

arctan .2 29 24 20 3 4 4 3 4 2

d xdx dx xI Cx x x x

+ + = = = = +

+ + + + + +


a) 4 23 5

2 10xI dx

x x

+=

+ + b) 5 24 1

6 9 4xI dx

x x

=

+ + c)

4

6 22

2 7x xI dx

x x

=

+ +

Hng dn gii:

a) ( ) ( )

4 2 2 2 2

3 174 1 4 13 5 3 174 44 42 10 2 10 2 10 2 10

x x dxx dxI dx dxx x x x x x x x

+ + ++= = = +

+ + + + + + + +

( ) ( )2

22 2

2

2 103 17 3 17ln 2 104 8 4 82 10 1 795

2 4 16

d x x dx dxx x

xx x x x

+ += + = + + +

+ + + + + +

( ) ( )2 2221

3 17 3 17 4 4 14ln 2 10 ln 2 10 . arctan .4 8 4 8 79 791 79

4 4

d xx

x x x x C

x

+ +

= + + + = + + + +

+ +

Vy ( )24 3 17 4 1ln 2 10 arctan .4 2 79 79xI x x C += + + + +

b) ( ) ( )

5 2 2 2 2

1 12 9 4 12 94 1 13 46 9 4 6 9 4 3 6 9 4 6 9 4

x x dxx dxI dx dx dxx x x x x x x x

+ +

= = =

+ + + + + + + +

( )( ) ( )

( )( ) ( )

22

2 22 2

6 9 4 3 11 1 44 ln 6 9 43 6 9 4 3 33 1 3 3 1 3

d x x d xdxdx x xx x x x

+ + += = + +

+ + + + + +

( ) ( )2 251 4 1 3 1 1 4 3 1ln 6 9 4 . arctan ln 6 9 4 arctan .3 3 33 3 3 3 3x x

x x C I x x C+ + = + + + = + + +

c) 4 3

2 26 2 2 2

2 25 7 2 25 72 4 1 232 7 2 7 2 7

x x x x xI dx x x dx x x dxx x x x x x

= = + + = + +

+ + + + + +

t ( ) ( )

2 2 2 2

25 2 2 32 2 225 7 252 3222 7 2 7 2 7 2 7

x x dxx dxJ dx dx dxx x x x x x x x

+ +

= = =

+ + + + + + + +

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( )( ) ( )

( )( ) ( )

22

2 2 22

2 7 125 2532 ln 2 7 322 22 7 1 6 1 6

d x x d xdxdx x xx x x x

+ + += = + +

+ + + + + +

( ) ( )32 2 2625 32 1 2 25 32 1ln 2 7 arctan 2 ln 2 7 arctan .2 3 26 6 6 6x x x

x x I x x x x C + ++ + = + + + + +

Tng kt: Qua ba phn trnh by v hm phn thc c mu s l bc hai, chng ta nhn thy im mu cht gii quyt bi ton l x l mu s.

Nu

( )( )

( )( )

21 2 2

1 2

22 22 2 2

222

( ) 1

( ) 1arctan

1

+ + = = +

+ +

+ + = + + = ++ + +

+ + = + = +

P x A Bax bx c a x x x x

a x x x xax bx cP x du u

ax bx c mx n k Cax bx c u

duax bx c mx n C

uu

BI TP LUYN TP:

7) 7 24 1

2 1xI dx

x x

=

+ + 8) 8 2

3 74 4 1

xI dxx x

+=

+ + 9)

2

9 23 1

9 6 1xI dx

x x

+=

+ +

10) 2

10 24 3 14 4 1

x xI dxx x

+=

+ 11)

2

11 22 3 2

4 4x xI dx

x x

+ +=

+ 12) 12 2

3 26 9

xI dxx x

=

+

13) 13 22 3

4 5xI dx

x x

=

+ 14) 14 2

3 12

xI dxx x

+=

+ + 15) 15 22 1

dxIx x

=

+

16) 16 22 1

4xI dx

x x

=

+ 17) 17 2

14 1

xI dxx x

+=

+ + 18) 18 2

4 11

xI dxx x

+=

+

III. MU S L A THC BC BA

Khi Q(x) = ax3 + bx2 + cx + d. Ta c bn kh nng xy ra vi Q(x). TH1: Q(x) = 0 c 3 nghim phn bit x1; x2; x3 Tng t nh trng hp mu s l bc hai c hai nghim phn bit. Ta c cch gii truyn thng l phn tch v ng nht h s. Ngoi ra ta cn c th s dng phng php bin i t s cha o hm ca mu (ty thuc vo biu thc ca t s l bc my) Ta c ( )( )( )3 2 1 2 3

1 2 3

( )( ) ax ( )P x A B CQ x bx cx d a x x x x x x Q x x x x x x x= + + + = = + +

ng nht h s hai v ta c A, B, C. Bi ton quy v nguyn hm c mu s l bc nht xt trn. Ch : vic ng nht c, th ta vn phi tun th nguyn tc l bin i sao cho bc ca t s phi nh hn bc ca mu s. V d 1. Tm nguyn hm ca cc hm s sau:

a) ( )( )1 22 9dxI

x x=

b) ( )2

2 2

6 21

x xI dxx x

+ =

c) ( )4 2

3 2

3 3 72

x x xI dxx x x

+ =

+

Hng dn gii:

a) ( )( ) ( )( )( )1 2 2 3 32 9dx dxI

x x xx x= =

+

Ta c ( )( )( )21 1 ( 9) ( 2)( 3) ( 2)( 3)

2 3 3 2 3 3A B C A x B x x C x x

x x x x x x= + + + + +

+ +

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150

10 530

1 9 6 6 16

AA B C

B C BA B C

C

=

= + +

= + =

= + =

Nhn xt: Ngoi cch gii truyn thng trn, chng ta c th bin i cch khc nh sau m khng mt nhiu thi gian cho vic tnh ton, suy ngh:

( )( )( ) ( )( )( ) ( )( ) ( )( )11 ( 3) ( 3) 1 1

2 3 3 6 2 3 3 6 2 3 6 2 3+

= = =

+ + + dx x x dx dxI dx dx

x x x x x x x x x x

n y, bi ton tr v cc dng bin i n gin xt n!

b) ( ) ( )( )2 2

2 2

6 2 6 21 11

x x x xI dx dxx x xx x

+ + = =

+

Cch 1: Ta c ( )( )2

2 26 2 6 2 ( 1) ( 1) ( 1)1 1 1 1

x x A B Cx x A x Bx x Cx x

x x x x x x

+ = + + + + + +

+ +

2

2 3 563 2 3 52 21 2ln ln 1 ln 1 .2 1 1 2 2

2 52

AA B CB C B I dx x x x C

x x xA

C

= = + +

= + = = + + = + + + + +

= =

Cch 2: ( )( ) ( ) ( ) ( )2 22

2 3 3 3 32

2 3 1 1 1 3 1 16 2 21

x x x dx x dxx x dxI dx dx dxx x x x x x x xx x

+ + +

= = = + + =

( )3 332 2ln( 1) ( 1)( 1)

d x x dx dxdx x x J Kx x x x xx x

= + + = + ++ +

Vi ( 1) 1 1 ln ln 1 ln( 1) ( 1) 1 1dx x x xJ dx dx x x

x x x x x x x

+ = = = = + = + + + +

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

dx x x dx dx x x x xK dx dx dxx x x x x x x x x x x x x x

+ + = = = = =

+ + + +

( 1) 1 ( 1) ( 1) 1 1 1 1 1 1 1 1ln ln( 1) 2 ( 1)( 1) 1 2 1 1 2 1x x x x x xdx dx dx dxx x x x x x x x x x

+ = = =

+ + +

T ta c 321 1 12ln ln ln ln .

1 2 1x x xI x x C

x x x

= + + ++ +

Nhn xt: Cch phn tch nh trn vn cha thc s ti u, cc em hy tm li gii khc thng minh hn nh!

c) ( ) ( )4 2 2 2

3 2 2

3 3 7 8 3 7 33 3 322 2

x x x x x xI dx x dx x Jx x x x x x

+ + = = + = +

+ +

Vi ( ) ( )( )2 2

2

8 3 7 8 3 71 22

x x x xJ dx dxx x xx x x

+ += =

++

Ta c ( )( )2

28 3 7 8 3 7 ( 1)( 2) ( 2) ( 1)1 2 1 2

x x A B Cx x A x x Bx x Cx x

x x x x x x

+= + + + + + + +

+ +

77 158 2 4 7 152 23 2 4 ln 4ln 1 ln 2 .

1 2 2 27 2 15

2

AA B CA B C B J dx x x x C

x x xA C

= = + +

= + = = + + = + + + +

+ = =

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Vy 2

33 7 153 ln 4ln 1 ln 2 .2 2 2xI x x x x C= + + + +

BI TP LUYN TP:

1) 1 2( 1)dxI

x x=

2) 2 22 1

( 1)( 9)xI dx

x x

+=

+ 3)

2

3 21

( 2)( 4 3)x xI dx

x x x

+ +=

+ + +

4) 4 25 2

(1 )(4 )xI dx

x x

+=

+ 5) 5 2

1( 4)xI dx

x x

+=

6) 2

6 2( 1)( 2)xI dx

x x=

+

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Nguyn tc gii: Khi bc ca t s P(x) ln hn Q(x) th ta phi chia a thc quy v nguyn hm c bc ca t s nh hn mu s. III. MU S L A THC BC BA

Khi Q(x) = ax3 + bx2 + cx + d. Ta c bn kh nng xy ra vi Q(x). TH1: Q(x) = 0 c 3 nghim phn bit x1; x2; x3 TH2: Q(x) = 0 c 2 nghim: mt nghim n, mt nghim kp Khi ta c ( )( )23 2 1 2( ) = + + + = Q x ax bx cx d a x x x x ng nht c, ta phi phn tch theo quy tc: ( )( ) ( )2 211 2 2

( ) ( )( )

P x P x A Bx CQ x x xa x x x x x x

+= = +

ng nht h s hai v ta c A, B, C. Bi ton tr v cc dng c bn xt n. Ch : Ngoi vic s dng ng nht, ta cng c th phn tch t s theo o hm ca mu gii. V d 1. Tm nguyn hm ca cc hm s sau:

a) ( )1 2 2dxI

x x=

+ b) ( ) ( )2 21

1 2 3xI dx

x x

=

+ c) ( )

2

3 22 4

2 1x xI dx

x x

+ +=

Hng dn gii:

a) Xt ( )1 2 2dxI

x x=

+

Cch 1: (ng nht hai v)

Ta c ( ) ( )( )2

2 2

140

1 11 Ax 2 0 22 42

1 2 12

AA B

A Bx C Bx C x B C Bxx x x C

C

=

= + +

= + + + + = + = ++

= =

Khi , ( )1 2 2 21 1 1

1 1 1 1 2 14 4 2 ln .2 4 2 4 2 4 22

xdx dx dx dx xI dx Cx x x x xx x x x

+ +

= = + = + = + + ++

Cch 2: (S dng k thut phn tch nhy tng lu ta c)

( )

( )( )

2 2

2 3 2 3 2 3 2 3 2 3 2

2 2

13 2 2 2 3 2 2

3 2

3 2

1 1 1 3 4 3 ( 2) 2 4 1 3 4 3 ( 2) 2 4.

4 42 2 2 2 2 2

1 3 4 3 2 1 3 4 3 14 4 4 22 2 2

21 3 1 1ln ln4 4 2 42

x x x x x x x x x x

x x x x x x x x x x x x

x x dx x x dx dxI dxx xx x x x x x x x

d x xdx x

xx x

+ + + + + + += = = + =

+ + + + + +

+ += + = = + =

+ + +

+= =

+

3 2 3 2

13 1 1 3 12 ln ln 2 ln .4 2 4 4 2

x x x C I x x x Cx x

+ + = + +

b) ( )( ) ( ) ( ) ( )2 2 21 2 21

2 51 2 1 5x tI d x dt

t tx x

+

= + = + +

, vi t = x + 1.


Ti liu bi ging:


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Ta c ( ) ( )( )2

2 2

1250 2

2 22 2 5 1 2 52 5 52 5

2 5 225

AA C

t At B Ct At B t Ct B A B

tt t tB

C

=

= + +

= + + + = =

= =

T ta c ( )( )

2 2 2 2

1 2 22 52 1 2 125 5 25

2 5 25 5 25 2 52 5

t d tt dt dtI dt dtt t tt t t t

+

= = + = + + =

1 2 1 1 2 5 2 1 2 3 2ln ln 2 5 ln ln .25 5 25 25 5 25 1 5( 1)

t xt t C C C

t t t x x

= + + = + = ++ +


( ) ( ) ( )( )

( )( ) ( ) ( )

( )2

2 2 2

2

3 2 2

56 10 3 2 5 2 52 5 22 1 2 1 4 2. .

2 5 5 2 5 252 5 2 5 2 5

1 1 2 4 6 10 3 5. .

5 2 5 25 2 5 2

t t t t tt tt

t t t tt t t t t t

t t

t t tt t t

= = =

=

( ) ( ) ( )3 222 3 2 2 3 2 2

2 52 5 2 51 1 4 6 10 12 2 7 1 4 25 5 2 5 25 25 5 25 5 2 5 25 52 5 2 5

d t td t d tdt t t dt dt dt dtI dtt t t t tt t t t t t

= + + + = + + =

3 22

7 1 4 2 1 1 2 1 2 5 2ln ln 2 5 ln 2 5 ln ln 2 5 ln .25 5 25 5 25 25 5 25 5

tt t t t C I t t C C

t t t t

= + + = + + = +

Thay li t = x + 1 ta c 21 2 3 2ln .25 1 5( 1)

xI Cx x

= ++ +

c) ( )2

3 22 4

2 1x xI dx

x x

+ +=


Ta c ( ) ( )( )2

2 22 2

2 2 92 4 Ax 2 4 Ax 2 1 1 2 4

2 12 1 4 20

A C Ax x B C

x x B x Cx A B Bxx x x

B C

= + = + + +

= + + + + + = + =

= =

( )2

3 2 2 22 4 9 4 20 20 49 4 9ln 10ln 2 1 .

2 1 2 12 1x x x dx dxI dx dx dx x x C

x x x xx x x x

+ + = = + = + = + + +


Ta c ( ) ( ) ( )( )

( )( ) ( )

( )2 22

2 2 2

6 2 6 2 12 1 22 4 2 1 4 2 4.2 1 2 1 2 1 2 12 1 2 1 2 1

x x x xx xx x

x x x x x xx x x x x x

+ + +

= + + = =

( ) ( )( )

2 2

33 2 2 2 2

3 2

6 2 6 22 1 2 24 4 1 28 44. 4.2 1 2 1 2 1 2 12 2 1

4 4ln 4ln 2 14ln 2 1 9ln 10ln 2 1 .

x x x xI dx

x x x x x xx x x x x x

x x x x C x x Cx x

= + + = + =

= + + + = + + +

TH3: Q(x) = 0 c 1 nghim n Khi ta c ( )( )3 2 21( ) = + + + = + +Q x ax bx cx d x x mx nx p , trong 2 0mx nx p+ + = v nghim. ng nht c, ta phi phn tch theo quy tc: ( )( ) 22 11

( ) ( )( )

P x P x A Bx CQ x x x mx nx px x mx nx p

+= = +

+ + + +

ng nht h s hai v ta c A, B, C. Bi ton tr v cc dng c bn xt n. Ch :

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- Nguyn hm = + +

2 2du 1 u

arctan C.u au a

- Ngoi vic s dng ng nht, ta cng c th phn tch t s theo o hm ca mu gii. V d 2. Tm nguyn hm ca cc hm s sau:

a) ( )1 2 1dxI

x x=

+ b) ( )( )2 2

2 31 4

xI dxx x

+=

+ c) ( )

2

3 2

11

x xI dxx x x

+=

+

Hng dn gii:

a) ( )1 2 1dxI

x x=

+


( ) ( ) ( )2

22

0 11 1 1 0 1

11 1 0

A B AA Bx C A x Bx C x C Bx xx x A C

= + = +

= + + + + = = ++

= =

Khi , ( )( ) ( )

22

1 2 2 22

11 1 1ln ln ln 1 .2 21 1 11

d xdx x dx xI dx dx x x x Cx xx x xx x

+

= = + = + = = + + + + + +


( )( )

( ) ( )2 2

212 22 2

11 1 1ln ln 1 .21 11 1

x x x dx xdxI dx x x Cx xx xx x x x

+ = = = = + +

+ ++ +

b) ( )( )2 22 31 4

xI dxx x

+=

+


( )( ) ( ) ( )( )2

22

350

2 3 32 3 4 1 21 541 4 3 4 7

5

AA B

x A Bx Cx A x Bx C x B C B

x xx x A CC

=

= + + +

= + + + + + = + = +

+ =

=

Khi ta c ( )( )2 2 2 222 3 3 1 3 7 3 3 7

. ln 15 1 5 5 5 54 4 41 4

x dx x x dx dxI dx xx x x xx x

+ += = + = + =

+ + + +

( ) ( )2 23 3 7 1 3 3 7ln 1 ln 4 . arctan ln 1 ln 4 arctan .5 10 5 2 2 5 10 10 2x xx x C x x C = + + + = + + + Cch 2: (S dng k thut phn tch nhy tng lu ta c) Ta c ( )( )

( )( ) ( ) ( ) ( ) ( )222 2 2

2 1 32 3 2 3 2 3; 1

2 51 4 2 5 2 51 1 2 1 5

xx tt x

t tx x t t t t t tx x x

++ += = = + =

+ + + + + + +

+ +

M ( )( ) ( )

( )2 2 2

3 2 22 2

3 4 3 2 5 23 1 1 3 4 3 2 1. . .

5 5 5 52 5 2 52 5 2 5

t t t t t t t

tt t t tt t t t t t

+ + + + += = +

+ + + ++ + + +

Suy ra

( )2 2

2 3 2 2 2 3 2 22

2 3 1 3 4 3 2 1 2 1 3 4 3 8 1. . . .

5 5 5 5 5 52 5 2 5 2 5 2 5 2 5 2 52 5t t t t

t tt t t t t t t t t t t tt t t

+ ++ = + + = +

+ + + + + + + + + + + ++ +

Thay vo ta c ( )( ) ( ) ( )2

2 3 2 22 2

2 3 2 3 1 3 4 3 8. .

5 5 52 51 4 2 5 1 4x t t t dt dtI dx dt dt

tt tx x t t t t

+ + += = = + =

+ + + + + + +

3 2 3 21 3 8 1 1 1 3 8 1 1ln 2 5 ln . arctan ln 2 5 ln . arctan .5 5 5 2 2 5 5 5 2 2

t tt t t C t t t C+ + = + + + + = + + + +

TH4: Q(x) = 0 c 1 nghim

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