TICH PHAN SO PHUC luyen thi Dai Hoc THPTQG

515.076
PH561P

1

iGV cbuyen Toan Trung tam luyen thi Vinh Viin - TP. HO Chi JViinhJ

PHUONG PHAP TINH


MATH-EDUCARE
HA VAN

CHUCING

(GV chuyen Toan Trung tarn luyen thi Vfnir Viin
TP. -Hd Chf Minh)

PHl/CfNG PHAP T I N H

TICK PHAN
VA so PHUfC
• LUYEN THI TU TAI VA DAI HOC
m

m

m

• CHirOfNG TRINH Mflfl NHAT CUA BO GIAO DUG VA DAO TAO
•

(Tdi

ban idn thii nhat,

IHU

VIEN

•

c6 siia chita

TiNH BIN'H

•

vd bo sung)

THUAN

NHA XUAT BAN DAI HOC QUOC GIA H A NOI

www.matheducare.com


MATH-EDUCARE
TICH PHAN
Hp N G U Y E N HAM
K I E N THLfC C d B A N

I. D i n h nghia
F(x) la nguyen hain cua fix) t r e n khoang (a; b) neu F'(x) = fix), Vx e (a; b)
k i hieu F(x) =

[f{x)dx .

II. Tinh chat
a)

(Jf(x)dxj' = f(x)

b)

[ f (x) ± g(x)]dx =

c)

kf(x)dx = k

f (x)dx ±

f(x)dx

g(x)dx

k e R

d) Neu F(t) la mot nguyen h a m ciia f(x) t h i F(u(x)) la mot nguyen

ham

ciia f [u(x)] u'(x).
I I I . B a n g n g u y e n h a m thi^dng d u n g v d i u = u ( x )
2.

.n + l
u"du = ^ — + C
n + l

= In u + C

4.

e"du = e" + C

cosudu = sinu + C

6.

sinudu = -cosu + C

du = u + C
3.

fdu
u

5.
7.

f

du
cos^ u

r

du
sin^ u

(1 + t a n u)du = t a n u + C
(1 + cot u)du = -cotu + C

du

1 , u - a
+ C.
= — I n
u^-a^
2a
u + a

T]

T i n h dao h a m cua F(x) = x.lnx - x, r o i suy ra nguyen h a m ciia f(x) = Inx.
Gidi
T a CO :

F(x) = x . l n x - x = x(lnx - 1)
3

www.matheducare.com


www.matheducare.com
Suy r a : F'(x) = [x(lnx - 1)]' = Inx - 1 + -

X

= Inx

Vay theo d i n h nghIa cua nguyen h a m , nguyen h a m ciia f(x) = Inx c h i n h
la F(x) = x l n x - x + C.
~2\h dao h a m ciia F(x) = x^lnx, r o i suy ra nguyen h a m ciia f(x) = 2xlnx.
Gidi
T a CO :
Vay

'

F(x) = x^lnx nen
F'(x)dx=

F'(x) = 2xlnx + — .x^ = 2xlnx + x = f(x) + x

f (x)dx +

dx

2

2

ff(x)dx = F(x) - — + C = x^lnx - — + C
J
2
2

=:>

V a y mo t nguyen h a m ciia f(x) la F(x) = x l l n x

.

~3\h nguyen h a m ciia f(x) - x V x + 1 biet F(0) = 2.
Gidi
Ta CO :

f(x) = x V x + 1 = (x + 1 - l ) V x + 1
=

(X + l ) V x + 1 - Vx + 1 = (X + 1)2 - (X + 1)2
3

Vay

f(x)dx =

1^

f{x + l ) 2 d x - ("(x + l ) 2 d x
3

1

(X + I ) 2 d ( x + 1 ) -

2

=

-

2

Hay
Vi

F(x)

f(X + l ) 2 d ( x + l )

-

- ( X + l ) 2 - - ( X + 1)2

5

+C

3

MATH-EDUCARE

- ( x + l ) V x + 1 - - ( x + 1) Vx + 1 + C
5
3

=

- ( x + l ) V x + l - - ( x + l)Vx + l + C

=

F(0) = 2 nen t a CO :

+ 1 - - (0 + 1)V0 + 1 + C

- (0 + 1)^ Vo
5

2=

5

3

3

15

F ( x ) = - ( x + D^Vx + l - - ( x + l ) V x + l + — .
5
3
15

V4y

4


MATH-EDUCARE
U 1

Cho F(x) = x h i x

va

1 . Chufng to r a n g :

f(x)

g(x) = x^ I n

=

; x > 0.

- g \ x ) - - X .

2

2

2. Suy r a mot nguyen h a m F(x) ciia f(x).
Gidi
1 . Ta CO :

g'(x) = 2 x l n

^x^

Suy r a :

2xhi

Vay

f(x) = - g ' ( x )
2

4)

(do

+ X

= g'(x) -

X >

0)

xhi

X

- - X
2

= igXx)-ix

(*)

2. Tix (*) t a suy r a :

J

f(x)dx = - ("g'(x)dx - 2 J
2

J

xdx = - g ' ( x ) - — x^ + C = - x^ h i^1
2
4
2
4.

Vay m o t nguyen h a m cua f(x) la : F(x) = - x I n

+ C

v4y

~5\. Chufng m i n h r a n g F(x) = 9 + (x - 2 ) 6 " la mot nguyen h a m ciia
{(x) = (x-

l)e\

2. Chufng m i n h r k n g G(x) = - ( 1 + x)e"'' la m o t nguyen h a m cua
g(x) = x.e'". Roi suy r a nguyen h a m cua k(x) = (x - De"".
Gidi
1.

Ta

CO

:

F'(x) = e" + e^Cx - 2 ) = e^lx - 1) = f(x)

V a y F(x) l a m o t nguyen h a m cua f(x).
2. Ta CO :

G(x) = - ( 1 + x)e"''

Suy r a : G'(x) = -e"" + (1 + x)e-'' = e"\ = g(x)
Vay G(x) la mot nguyen h a m ciia g(x).

Suy r a nguyen h a m cua k(x) = (x - l)e~'' = xe"" - e"'' = g(x) - e"
Nen

k(x)dx = Jg(x)dx - je-^dx = G(x) + e"" + C
=

- ( 1 + x)e-'' + e " + C = - x e " + C

V a y nguyen h a m cua k(x) \k K(x) = -xe"" + C.

www.matheducare.com

5


www.matheducare.com
~G\h dao hkm cua (p(x) = (ax + b)e''. Roi suy ra nguyen ham cua fix) = -xe".
Gidi
(p'(x) = (a + ax + b)e''

Suy ra :

(p(x) = (ax + b)e''

TiX gia thiet

De tinh nguyen ham ciia f(x) = -xe" ta chon a = - 1 , b = 1
Thi

=>

(p'(x) = -xe"

Vay nguyen ham ciia f\x) = -xe" la F(x) = (-x + De" + C.
~T\g minh F(x) = In| x + Vx^ + K | la mot nguyen ham cua
fix) =

, ^
Vx^+K

tren R.

Gidi
^
,
i a CO :

1+

.
(X + V 7 7 K ) '
r (x) =
X + Vx^ + K

Suy ra : F'(x) =

, ^

X

VX2 + K

x + Vx^ + K

VX^ + K +
Vx^ + K ( x + Vx^ + K )

= f(x) Vx e R.

Vx2+K

Do do : F(x) la mot nguyen ham cua f(x) tren R.

sl

Cho ham so fix) = xV3 - x vdi x < 3.
Tim cac so' a, b sao cho ham so' F(x) = (ax^ + bx + c) V3 - x la mot
nguyen ham cua f(x).
Dai hoc Su pham Ki thudt

TP.HCM

Gidi

MATH-EDUCARE
Ta CO :

F'(x) = (2ax + b) V 3 - x

(ax^ + bx + c).
2V3-X

1
2V3-X

.

.

.2

[(2ax + b).2(3-x) - (ax' + bx + c)]

1 . [-5ax^ + (12a - 3b)x + 6b - c)]
2A/3-X
F(x) \k mot nguyen hkm cua f(x) k h i x < 3
-5ax^ + (12a - 3b)x + 6b - c = 2(3 - x)x

o

F'(x) = f(x)

o

Vx < 3
Vx < 3

6

MATH-EDUCARE
2
a = 5

-5a = -2
12a - 3b = 6

b = -^
5

6b - c = 0

c - - —

el

Chvlng m i n h F(x) = — I n X - a v 6 i a > 0 l a m o t nguyen h a m ciia
x + a
2a
1
x^-a^

f(x) =

v d i Vx ^ ± a.

Gidi
r

^x-a^
T a CO :

F'(x) =

1

^x + a j

F'(x) =

1

2a' rx-a^

U

Suy r a : F'(x) =

2a
af

(X +

Vx

^x-a^

2a'

9^

± a.

vx + a.j

+ a,
2a

+a

X

2a ( x + a f

Vx ;t ± a

x-a

1

1

(x + a)(x - a)

- a^

= flx)

V a y F'(x) l a m o t nguyen h a m cua f(x) v d i Vx ^ ± a.

10 1 ChiJng m i n h

F(x) = •

neu x ^ O
X
1

neu x = 0

(x-De^+l
x^

f(x) = <
-

I

I V
l a nguyen h a m cua

neu

x^O

neu x = 0

2

Gidi

* Khix^Othi

F'(x) =

e^.x-Ce"-l).l

(x-De^+l

X

= f(x)

X

V a y F(x) l a nguyen h a m cua f(x) t r e n (-oo, 0) u (0, + « )
e" - 1
* K h i x = 0 t h i F'(0) = l i m ^^^^
x-^'O

^^^^ = l i m

X - 0

(1)

-1

x

7

www.matheducare.com


www.matheducare.com
F'(0) = l i m
X-.0

Vay

^

^ = lim
x^o

FXO) = l i m — = - = fTO)
x^o 2
2

2x

^

(Quy t^c L'Hopital)

(2)

TiT (1), (2), t a suy r a F(x) l a nguyen h a m cua f(x) t r e n R.
IT]

T i n h dao h a m cua F(x) = (x^ - l ) l n 1 1 + x | - x^ln | x i .
Suy r a nguyen h a m cua f(x) = x l n

1+x

A2

Gidi

Ta

CO

:

Suy r a

Taco:

F(x) = (x^ - l ) l n

11

+ x | - x^ln | x I
x^-1

F'(x) = 2 x l n I x + 1 1 +

x +

1

- 2xln I X I

-

= 2 x l n I X + 1 i - 1 - 2xln | x | v 6 i x
0, x ^ 1
2
f l + x^2
+
x
+
x
il
= xln
= 2xln
f(x) = x l n l
X
X

1

h

x

1

i - I n 1X1 ]= 2 x l n 1 1

+

X

f(x) = F'(x) + 1

Tir ( * ) , ( * * ) t a suy r a :
Suy r a

1-

(**)

ff(x)dx = [F '(x)dx + f l d x = F(x) + x + C

V a y nguyen h a m cua f(x) = x l n I

1

+

X

1

la :

F(x) = ( x 2 - D l n l l + x | - x ^ l n l x l + x + C.

12 I T i m a. b, c sao cho F(x) = e""^ (atan^x + btanx + c) l a m o t nguyen h a m

n n

cua f(x) = e'"^ .tan^x t r e n

MATH-EDUCARE

I

2'2

Gidi

Taco :

F'(x) = 7 2 . 6 " ^ ( a t a n ^ x + b t a n x + c) + e ' ' ^ [ 2 a ( l + tan^ x ) t a n x + b ( l + tan^ x)]
v d i Vx e
7t

F(x) l a nguyen h a m cua f(x) t r e n

71

{'2'

2)

<=> F'(x) = f i x ) , Vx

7t _ 71

2' 2.

n

71

'2' 2

8


MATH-EDUCARE
2atan^ x + (yl2a + b ) t a n 2 x + (^^b + 2 a ) t a n x + (V2c + b) = e ' ' ^ t a n ^ x

<=> e

Vx e

2' 2

1
a = 2

2a = 1
V2a + b = 0

b = -

A ^ b + 2a = 0

2

.

1
c =—
2

V2c + b = 0

13 I Chutog m i i i h F(x) = | x | - l n ( l + I x I ) la mot nguyen ham cua fix) 1+
Dai hoc Tong

hap TP.HCM

-

1993

Gidi
x - l n ( l + x)
Ta

CO

:

F(x) =

neu x > 0

0

neu

- x - l n ( l - x)

0

X =

neu x < 0

neu x > 0
Ta

CO

:

f(x) =

1 + x
0

neu x = 0

V

1-x
Do do, t a
* Khi

neu

CO :

0 thi

F'(x)

= 1 -

* Khi x < 0 thi

F'(x)

= -1 +

* vu* Ehi

X >

X =

n.u^
0 thi

1+x

1+x

1-x

= f(x)

1-x

(1)

= f(x)

(2)

l^vn+^ r
F(x) - F(0>
,.
x - l n ( l + x)
F (0 ) = l i m
= lim
X - 0

x^O"

= lim

x-»0*

F'(0*) = 1 - l i m
x^.0^

F'(O^) = 1 - 1

x^O"

X

,.
l n ( l + x)
lim

X

x-»0* X

Suy ra :

0

X <

^—
1+ X

x

(do quy t i c L ' H o p i t a l )

= 0

www.matheducare.com


www.matheducare.com
TV/r-.iu'

v

F(x)

-

F(0)

Mat khac : F (0 ) = l i m

,.

- X - ln(l

-

x)

= lim
x - 0

x-»o-

X

FXO-)=-l-liml^^^
x->0"

X

-1

^

F'iOl

= - 1 - l i m -i-tiL = - 1 + 1 = 0
1

x-»0-

Vay

F'(0*) = F'(0") = 0

(3)

F'(0) = 0 = flO)

Nen

TCr (1), (2), (3) suy ra F(x) la nguyen ham ciia f(x) tren R.

14 I Churngminh

F(x) =

— In x

(X

2

4

•

0

fx In X
f(x)

>0)

la nguyen ham cua

(x = 0)

(x > 0)
(x = 0)

0

Dai hoc Yduac

TP.HCM

Gidi

K h i X > 0 ta CO : F'(x) = x l n x +

2

2

X

X

(1)

= xlnx = f(x)
x^,
In

Khi

0 ta

X =

CO

X

-

: F(O^) = l i m ^^""^ ^^^^ = l i m
X - 0

x^O*

= l i m — In

X

x-yO* 4

x^

X

-0
2

x^.0*

(quy t^c L'Hopital)

2

x^o-

^

x ^ O *

- lim — = lim

2

x-^O*

= lim

MATH-EDUCARE
= lim

= 0 = f(x)

x^o^V

(2)

2)

Tix (1), (2) ta ket luan F(x) la nguyerI hkm cua fix) tren [0, +oo).
15 i Tinh dao h^m cua ham so F(x) = In

fx'-2^

+ 1]

^x^ + 2^R + lj
x^-1

Roi suy ra ho nguyen ham cua f(x) =

xUl

10


MATH-EDUCARE
Gidi
Ta CO :

F(x) xac d i n h vdfi m o i x.
x^ - 2^

Ta

CO

:

F'(x) =

+1
2V2(x2 - 1)

x^ + 2 A / X + 1

x^ + 2V^ + 1

(x^ + 2V^ + 1)2 ' x^ - 2 A / i + 1
x^ + 2Vx + 1
2V2(x2 - 1)

2^{x'^ - 1)

= 2V2f(x)

x^ + 1
f(x)dx =

Vay

x^-l
x^+1
16

x^ - 2 7 ^ + 1^
^ : F ( x ) = i l n
2A/2
2V2
x^ + 2A/X + 1

dx =

In

fx2-2>^ + l

x^ + 2Vx + 1

T i m ho nguyen h a m cua f(x) = max ( 1 , x^).
Gidi
Ta

CO

:

f i x ) = max ( 1 , x^) =

'x^ neu
1

X <

neu - 1

- 1 V 1
< X <

< X

1

Vay:
j x ^ d x neu x < - 1 v 1 < x

+ C neu

m a x ( l , x2)dx =
Idx

neu - 1 < x < 1

17 I T i m ho nguyen h a m ciia f i x ) = | l + x| -

X

+C

x < - l v l < x

neu - 1

< X <

1

|l-x|.

Gidi

Ta

CO :

vay

1 +x -

j ;1 + x -

l +x

dx =

-2dx

neu x < - 1

2xdx

neu - 1 < x < 1

2dx

neu x > 1

-2x + C vdi
l +x

dx = x^ + C
2x + C

X <

-1

vdi - 1

< X <

vdi

1

X >

1

11

www.matheducare.com


www.matheducare.com
18 I T i m ho nguyen h a m ciia f(x) = | x |
Gidi

Ta CO :

neu x < 0

-xdx

neu x > 0

xdx

+C

neu X > 0

I X I dx =
+ C neu X < 0
2

19 I T i m ho nguyen h a m ciia f(x) = x i x |.
Gidi

Ta CO :

20

neu x < 0

-x^dx

neu x > 0

x^dx
J x I x I dx =

T i m ho nguyen h a m ciia f(x) = (x +

— +C
3

neu X > 0

x^
+ C neu x < 0

\x\f.

Gidi
Ta CO :

(x+ I X I )2dx = Jlx^ + 2x I X I +x2 )dx
'

4x

2

4x dx =

+C

O.dx = C

21 I T i m ho nguyen h a m ciia f(x) =

neu x > 0
neu X <

0

cos X

Dai hoc Yduac

TP.HCM

-2001

-He

nhdn

Gidi

MATH-EDUCARE
T a CO :

F(x)

' d(sin x)

•COS xdx

= f
• cosx

• d(sin x)
sin^ X - 1

1 - sin^ X

COS'^ X

1,
sin X - 1
+ C.
-In
2
22I

I sin X + 1

T i m ho nguyen h a m cua fix) = 2\3^\5^\
Gidi
Ta CO :

f(x)dx =

2 \ 3 2 \ 5 3 M X = f(2.32.5^)''dx =

2250"
hi(2250)

+ C.

12


MATH-EDUCARE
23 I Tim ho nguyen ham cua :

X* +

a) f(x) =

X

2x^

+ X +

2

b) g(x) =

+x+1

x^ + x^ + 1
X^ + X +

1

Dai hoc Ngoqi thuang - 1998
Gidi
f(x)dx =

a)

x^ + 2x2
X

+ X +

+ X +

2

1

x^ + x^
1
fx''
x" + X
g(x)dx = —
dx=
J x^ + X + 1

b)

dx =

(x^ -

X +

x^
2)dx = 3

C o

x"^

X^

3

2

( x ^ - x + l)dx =
J

(hi

24 I Tim ho nguyen ham cua f(x) =

2 + 2x + C
+ X + C.

X)'

Gidi
Ta

f (x)dx =

CO :

f(lnx)4

dx = j ( h i x)"* d(ln x) = - (In x)^ + C.

25 \m ho nguyen ham cua f(x) =

——
e" - 4e "

DH Quoc gia Ha Ngi - D/1999

Gidi
Ta

f(x)dx =

CO :

f e^dx

Tim ho nguyen ham cua f(x) =

d(e'')

e" - 2
=lln
+ C,
4
6^+2

e^^
e" + 1
Gidi

Ta

CO :

f (x)dx =

'(e" + IKe^" - e" + 1)
^-^dx
+1

I 27 I Tim ho nguyen ham cua f(x) =

=

1-e

f ( e 2 ' ' - e ' ' + l ) d x = i e 2 ' ' - e ^ + x + C.
J
2
2x

Gidi
Ta
f(x) =

CO :

f eMx
1-e

2x

d(e'')

die"")
2

e" - 1 + C = - h i 6=^+1
+C
2
e" + 1
e" - 1

www.matheducare.com

13


www.matheducare.com
2 8 ] T i m ho nguyen h a m cua fix) = Ve" + e

- 2.
DH Y Thai Binh

-

1997

Gidi
Ta

CO

f(x) = Ve" + e"" - 2 =

-e"2

e2

X

X

f(x) =

X

e2 - e 2

.

neu

'

X

X

—>

2

e2 - e 2

—
2

X

.

'

X

X

X >

neu

e2 - e'2

—<
2

neu

-e2 + e 2
X

f(x) =

—
2

X

X

0

X

-e2 + e 2

r

neu

0

X <

xA

X

e2 + e 2

Nen

X <

neu

X >

neu

0

f (x)dx =
e2 + e 2

29 I T i m ho nguyen h a m ciia f(x) -

0

Inex
1 + xlnx
HV

Quan he Quoc te -

1997

Gidi
T a CO :

d ( l + x l n x ) = (1 + xlnx)'dx = ( l . l n x + — .x)dx
X

= (Inx + l ) d x = Inex.dx

MATH-EDUCARE
Vay

f(x)dx =

1 + xlnx

1 + xlnx

rd(l + xlnx)

Inex.dx

f (x)dx = I n 1 + x l n x

+ C.

3o] T i m ho nguyen h a m cua f(x) = x ( l - x)-°.
DH

Quoc gia Ha Ngi

-

1998

Gidi
Ta

CO

: fix) = [(x - 1) +1](1 - x)^" = (x - 1)^' + (x - 1)^°.

14


MATH-EDUCARE
f(x)dx=

Nen

(x-l)2Mx +

(x-l)^M(x-l) +

(x-l)2°dx

(x-l)2°d(x-l) =

(X-1)22

(x-l)21

22

21

+ C.

,2001

31 I Tim ho nguyen ham cua f(x) =

(1

+

X2)1002

DH Quoc gia Ha Ngi - 2000
Gidi
.2001

Ta

CO

:

f(x) =

,

( l + x^r^^

f

2000

( l + x ^ r o ' d + x^f
/-

.2001

J f (x)dx =
(1^^2)1002
2

a) Dat x = tana.

X

1000

NIOOO

X

•(l + x^)^

\1000

dx

2

1

^

1 + x^

1 + x'

^

1 + x^

1 + x^
^

32 1 Tinh tich phan

,

'

dx =

..2

2002 1

+ C.
+

X ^

x^dx
—
bang hai each bien ddi sau :
(x^ + if

b) Dat u = x^ + 1 So sanh hai ket qua t i m ducfc.
Dai hoc Tong hap TP.HCM

~ A/1977

Gidi
Ta

CO

a) Dat

:

X = tana
II

=

=>

dx = (tan^a + l)da,

x^^dx

rtan^ a(tan^ a + l)da

(x^ +1)^

(tan^ a +1)^

tan^ ada

rsin^ a

(tan^ a +1)^

cos a

sin^ a. cos ada =
=

thi :

da

(vi tan^a + 1 = — ^ — )
cos a

cos'' a

sin ad(sin a)

— sin* a + C = - (tan* a. cos* a) + C
4
4
tan* a
1
x^
1
+ C.
+
C
=
-.4 • (tan^ a + lf
4 ' (x^ + 1)^

www.matheducare.com

15


www.matheducare.com
b)

Dat
I,

u =
r

+ 1

x^dx

=>

du = 2xdx

• x^.xdx

(x^+l)^ ~ J (x^+l)^
-2
-3o..
(u"" -u-')du
=

1 r(u - D d u
'2 .

1 . 1 .
l - 2 u , ^
- l ( l + 2x^)
+ — + Ci =
— + Ci = —
+ C,
4u^
4 (x^ + 1)2
'
4u^

2u

T a xet : I i - I2 =
4(x2 + 1)2
I i - I2 =

1 (1 + 2x2)
+ C+ - C,
4 ( l + x2)2

x^ + 2 x ^ + 1
4(x2 + i f

+ C - Ci

I i - I2 - - + C - C i .
4
33 I T i m ho nguyen h a m cua fix) =

Vx^ + x"" + 2

Gidi
Ta

CO :

Vx^ +x-^

f(x)dx =
I x2

+ X-2

I

X

+2

2

+ X

r 1

\

,

Biet

V(x2+X-2)2

x - ^ ) d x

dx

-2

dx

(do x^ + x"2

= X

2 + -1- > 0)

1

X-*

+

=hilxl - —
4

X

34

+ C=

= ln(x.Vx2 + 3 ) + C . T i n h F(x) =

Inixl 4x^

-1999

Gidi

= [Vx2 + 3dx = xVx2 + 3 - jx.d(Vx2 + 3)

Fix)

= x Vx^ + 3 -

+C.

fVx2 + 3dx .

DHYHaNoi

MATH-EDUCARE
^•^'^^
> ^ 3

= x V 7 T i -

(Tich p h a n t i i n g phan)

f V 7 7 ^ d x + 3'"

4.x^ + 3

F(x) = xVx2 + 3 - F(x) + 31n(xVx2 + 3 ) + C
F(x) = i x V x 2 + 3 + - ln(xV(x2 + 3) + C.
2
2

Vay

16


MATH-EDUCARE
35 I T i m ho nguyen h a m cua f i x ) =

1 + 8"
HVNgdn

hang

-2000

Gidi
Ta

CO

:

dx

F(x) =

1 + 8" - 8"
dx =
1 + 8"

1 + 8"

Vay

du
In 8

= x -

8"dx
1 + 8"

= 8"dx.

f i i i L =J _ i n ( l ^ 8 " ) + C
J u In 8
In 8

_!L^x=
1 + 8"
dx

F(x) =

1 + 8"

du = 8".ln8.dx

D a t u = 1 + 8"

Nen

'S^dx

Idx

1 + 8"

=

X -

l n ( l + 8") + C.

hi 8

3 6 ) T i m ho nguyen h a m cua f(x) =
Vx^ - x - 1
DH Y Thai

Binh

Gidi
D a t t = V u ^ + K + u => d t = :

du

+ 1 du

dt

Ap dung :
df

dx

F(x) =

Vx^ - x - 1

d

1^
X

- f

i

fx

r

^
.

F(x) = I n Vx^ -

X -

1

+ X -

X

f
1
2)

5
4

'

5

2J

4

+ X

+ X

1

—

2

1

—

2

+ C.

-

(x + 3)^
37 \m ho n g u y e n h a m cua f i x ) =
(x - 7)^
Gidi
Dat u = x - 7
Vay

F(x) =

du = dx,

u + 10 = x + 3

(x + 3 ) ' ^ ^ ^ j-(u_+10)'
(x-7)^

^

u^

du

THLT VIcNTIf^HBiNHTHUAN

www.matheducare.com

17


www.matheducare.com
5-k

k=0
5

k =0

k =0
-1-k

"

T i m ho nguyen h a m ciia f(x) =

5

k=0

-1-k

-1-k

+ c.

x^-l
(x^ + 5x + IXx^ - 3x + 1)'
DH Quoc gia Hd Ngi - A/2001

Giai
Ta

CO :

F(x)=

x^-l

2x + 5
1
2x-3
• + - .+ 5x + 1 8 • - 3x + 1

1

(x^ + 5x + IXx^ - 3x + 1)
- 8

X

2x + 5

J
1
•dx + 5x + 1
+

39 i T i m ho nguyen h a m cua f(x) =

X

2x - 3
,
1 , x-" - 3x + 1
-dx = - I n
3x + 1
X
+ 5x + 1

+ C.

-

x l n X. ln(ln x)

Gidi
Ta CO : (hi(lnx))' =

Nen

xlnx
(•[In(lnx)]'

dx

f(x)dx =

x l n x . h i ( l n x)

Ta

CO :

h i ( l n x)

= In(lnx) + C.

(x + 1)

40 I T i m ho nguyen h a m cua f(x) =

xQ + xe")

Gidi

MATH-EDUCARE

(1 + xe")' = e^Cx + 1)
f (x)dx =

xe^Cl + xe")

x d + xe'')

e^Cx + Ddx

(x + Ddx

dCl + xe")
[1 + xe^ - I J U + x e " ]

(1 + xe" - 1)

[ l + x e " - l ] [ l + xe'']

r d ( l + xe^ - 1)

[ ( l + x e ' ' ) - ( l + x e ' ' - D l d C l + xe")

= hij 1 + xe^ - l l - Inl 1 + xe" I + C = In

xe
1 + xe^

r d d + xe")
d + xe")

+ C.

18

MATH-EDUCARE
41 I T i m ho nguyen h a m cua f(x) -

2x
x +Vx^ - 1
Gidi

Ta

CO :

f(x)dx =

2xdx

f 2x(x - Vx^ - 1)

x +Vx^ - 1

x^ - (x^ - 1)

2x2dx -

3

2xVx2 - I d x = - x^ 3

dx

(X^ -l)2d(x2

-1)

3

42 I T i m ho nguyen h a m ciia f(x) =

Vx + 3 + Vx + 1
Gidi

Ta

CO :

Vx + 3 - Vx - 1
dx
J(x + 3 ) - ( x + l)

dx

f{x)dx =

Vx + 3 + Vx + 1

i
1 f
i
(x + 3)2d(x + 3 ) - - (x + l)2d(x + l )
2J

= - V ( x + 3)=* - - V ( x + 1)^
3
3

+C.

3x + l
43 I 1. Xac d i n h cac h^ng so' A, B sao cho

(X +

1)^

A

B

(x + 1)^

(x + 1)^

2. Dua vao k e t qua t r e n , t a t i m ho nguyen h a m cua f(x) =

3x + l
(X +

ir

Gidi
1. Ta

CO

Vay
2.

B

Bx + (A + B)

(x + if

(x + If

3x + l
(x + 1)^

(x + if
B = 3

B = 3

A + B =1

A = -2

3x + 1

-2

3

(x + 1)^

(x + 1)^

(x + \f

3x + l

f(x)dx

(X +

If


dx
(X +

If

dx

+ 3

(x + If

= - 2 (x + ir^dx + 3 (x + ir'^dj^

=

(X+ i r - 3 ( x + i r + c =

(x + lf

X +

1

+ c.
19

www.matheducare.com


www.matheducare.com
44] Cho h k m so f(x) =

3x^ + 3x + 3
x^ - 3x + 2 '

1. Xac d i n h cac h k n g so A, B, C de fix) =

A
(x - 1)2

B
+

C
+•
x +2

x- 1

2. T i m nguyen h a m cua f(x).
DHYDuac

TP.HCM - 1996

Gidi
1. T a

CO

:

Do do

x^ - 3x + 2 = (x - l)^(x + 2)
f(x) =

o

(x - 1)2

x^ - 3x + 2

A

Bx^ + 3x + 3

+

B
X

-1

C
+•
x +2

3x^ + 3x + 3 = (B + Ox^ + (A + B - 2C)x + (2A - 2B + C)
A =3

B +C =3

2.

f (x)dx =

C =1

2A - 2B + C = 3

B =2

A + B - 2C = 3

3x2 + 3x + 3
x^ - 3x + 2

-3
x-1

+ 21n|

X

Ux-1)2

- l l + Inl X

45 1 T i m ho nguyen h a m cua f(x) =

+

• +

x+2

x-1

1

2
+ •

ix

2| + C.

.
x"^ + x^
DH Dagc Ha Ngi - 1997
Gidi

Ta

CO :

fix) =

(l + x2)-x2
X ^ d + X^)

x^+x^

MATH-EDUCARE
.
1
fix) = —

( l+ x2)-x2

X^

Vay

X(l+

x^

fdx

x2)

•dx

fdx

f (x)dx =
fdx

1

J

X^

X

X^

1

1

X(l +

+•
l +

x2)

x2

• xdx
X2+1

X

d(x2 +1)

= J _ - I n l x l + i l n ( x 2 + 1) +

x2+l

2x2

2

x +1
46 I T i m hp nguyen h a m cua f(x) = —
x^ + 4x^ + 4 x 2 - 4
20

u


MATH-EDUCARE
Gidi

Ta

CO

: f(x) =

x+1
x"* + 4x^ + 4 x ^ - 4

x+1
(x^ + 2xf - 4

x+1

(x^ + 2x + 2 ) ( x 2 + 2x - 2)
If
2x + 2
2x + 2
8 Kx'^+2x-2
x ^ + 2 x + 2.
Suy ra : F(x) =

x+1
x^ + 4x^ + 4 x ^ - 4

x+1

if

+ 2x - 2 x^ + 2x + 2>

2x + 2
2x + 2 dx
x^ + 2x - 2 x^ + 2x + 2

x'' + 2x - 2 + C.
x^ + 2x + 2
47 I Tim ho nguyen h a m cua f(x) =

Ta

CO :

X - X

x^ + 4x^ + 4 x ^ + 1

Gidi
x^-x
f (x)dx =
dx
x^ + 4x^ + 4 x ^ + 1
X

Dat u =

X+

— , ta

x'^ + — + 4
x^

J

1^

dx =

X+ —
Xy

1

X+ —

V
\

X+ —
X
X + —
X

CO :

X

f (x)dx = r du
u^ + u

u ^ + l - u ^

du = fdu

udu

u(u'' + 1)
.2
,
I I
1 fd(u' +1) , 1 1
1, , 2
^
= In I u I -2- J —
:= In I u I ln(u + 1) + C
^ X + —1
u^ + 1
2
Xy
+ C = - l n x^ + 2 x ^ + 1 + C.
2
2 x^ + 3 x ^ + 1
X+ i +1

u ^ + l

X/

48 I Tim hp nguyen h a m cua f(x) =

x' + 3 x ^ + 2

www.matheducare.com

21


www.matheducare.com
Gidi
Ta

CO :

f(x)dx =

(x^ + IXx^ + 2)

x^ + 3 x ^ + 2

[(x^ + 2) - (x^ + 1)

xdx

xdx

x^+l

r xdx
x^ + 2

1 fd(x2 + 1)
2J

x^+l

xdx

1 rd(x^ + 2)
2 J

x^ + 2

x^+1^
= -InCx^ + l ) - i l n ( x 2 +2) + C = - l n
+ C.
2
2
2
x^ + 2

49 I T i m ho nguyen h a m cua f(x) = 2 - 3 x '
Gidi
Ta

CO :

f(x)dx =

dx
2-3x2

r

dx

dx

3

3
1
3 J

x^ -

3-276

^Vef

1

dx

3x-Ve

pin
3x
2V6
50 i T i m ho nguyen h a m cua fix) =

1

X

-

hi

+ c
X +

+ c.

+Ve

X*

- 1

x(x^ - 5)(x^ - 5x + 1)
Gidi

Ta

CO :

MATH-EDUCARE
"f(x)dx =

'(x^ - Ddx
x'' - 5 x

f [(x^ - 5x + 1) - (x^ - 5x)](x'' - Ddx

(x* - l ) d x

x(x'' - 5)(x^ - 5x + 1)
r (x"* - Ddx

(x^ - 5x)(x^ - 5x + 1)

1 f d(x^ - 5x)

(x^ - 5x + D

5J

1 fd(x^ - 5 x + D
x^ - 5x + 1

x" - 5 x

x^ - 5 x
= i I n I x*^ - 5x I - i I n I x^- 5x + 11 + C = - I n
+ C.

5
5
5
x^ - 5x + 1

5 1 I T i m ho nguyen h a m cua f(x) =

x

+X

+

1

- D^

(X


MATH-EDUCARE
Gidi
Ta

CO :

X

(x-lf
x^ +

B

1

+ X +

(x-lf

x-1

(x-lf

1 = A + B(x + 1) + C(x -

X +

i f

Dong n h a t => A = 3, B = 3, C = 1.
X

1

+ X +
(X

(x - if

-1)^

f(x)dx =

X

(X -

+ X +

1

(x -1)-*

dx

dx = 3

-3
2(x - I f

x-1

1)2

(x - if

r

dx

+ 3

(X -

dx
x-1

If

+ I n ! x - 11 + C.

x-1

52 I T i m ho nguyen h a m cua f(x) = sin^x.cosx.
Gidi
Ta

CO :

f (x)dx =

sin^

X COS

. 3

xdx =

sm

x.d(sm x) =

sin^

X

+ C

'x^
cos
.2.
53 I T i m ho nguyen h a m ciia f(x) = •
X + sin x
Gidi
cos
Ta

CO :

f (x)dx =
X

v2y

+ sin

X

r l + cosX ,

1 f d ( x + s i n x)
dx = X + sin X
X + sin X
2J

dx = l
2

= - ln(x + sinx) + C = I n Vx + sin x + C ( v i x > sinx).
2
54 I T i m hp nguyen h a m cua f(x) = -7^
sin X
Gidi
Ta

CO :

f (x)dx =

dx
sin

X

f s i n xdx

f s i n xdx

sin^

1 - cos

-d(cos x)
cos X - 1

X

r d(cos x)

X

1, cos x - 1
= - In
cos X + 1
cos^ x - 1
2

+ c.
23

www.matheducare.com


www.matheducare.com
55 I T i m ho nguyen ham cua f(x) = tanx.
Gidi
Ta

CO

:

Jf (x)dx =

f-d(cosx)
f
tan xdx = rsm X ,
= - I n I cosx + C.
cosx dx = J cos x

56 I Tim ho nguyen ham cua ham so f(x) =

cos X + s i n x cos x
2 + sin X

Gidi
Ta

CO

:

f(x) =

cos x + sin X cos x _ 2 cos x + sin x cos x - cos x
2 + sin x
2 + sin x
cos x(2 + sin x) - cos x
2 + sin X

F(x) =

fcos x +sin X cosx
2 + sin X

= cos

dx =

X -

cos x
2 + sin X

cos X

V

COSX
-

2 + sin

X,

dx

= sinx - ln(2 + sinx) + C.
57 I Tim ho nguyen ham cua ham so f(x) =

sin 3x. sin 4x
tan X + cot2x

Gidi
Ta

CO

:

sin3x.sin4x _ sinSx. sin4x _ sinSx. sin4x
f(x) =:
tan X + cot2x
sin x cos 2x
cos x
+•
sin 2x
cos x. sin 2x
cos

X

= sin2x.sin3xsin4x = - [cosx - cos5x]sin4x
2
= — [sin5x + sinSx - sin9x + sinx]
4

MATH-EDUCARE
Vay

F(x)=

f (x)dx =

1
4

J

f sin3xsin4x
tan

X

+ cot 2x

dx

(sin 5x + sin 3x - sin 9x + sin x)dx

1
- — cos 5x - — cos 3x + — cos 9x - cos x+ C.
5
3
9
4
Tim ho nguyen ham cua ham so' f(x)

(sin''x + cos'*x)(sin®x + cos^x).
HV Quan he Quoc te -D/1997

24


MATH-EDUCARE
Gidi
Ta

CO : f(x) =

1

1-

9

- sin^

2

2x

\

3
1- -

5

2x

sin^

= 1

4

5 1
= 1 — . - (1 - cos
4 2

3
s i n ^ 2x + - sin"* 2x

3

8

n2

4x) + — — (1 - cos4x)
8 2

5
3
= 1 — (1 - cos 4x) + — [1- 2cos4x + cos^4x]
8
32
15
7

^
3
„ , 33
7
^
3
= — + — cos 4x + — (1 + cos 8x) = — + — cos 4x + — cos 8x
32 16
64
64 16
64
F(x)

=

33

f(x)dx =

7

—

—

+ — cos

. 64
.

7

33

^

163

64

64

4x +

.

— s i n 4x +

X +

3

— cos 8x

^

64^

8x

+ C.

sin

dx

512

5 9 I T i m h o n g u y e n h a m c i i a f[x) = t a n x

V2x + 1 + V2x -

DH Kinh

te Quoc

1 '

dan Ha Ngi

-

A/1999

Gidi
HGu

ti hoa ta c6 f(x) = tgx +

tan xdx =

fflUii dx

J cos

=

X

-d(cos x)
J

cos

V2x

+ Idx =

(2x + l ) 2 d x =

TA

(2X - l)V2x - 1

V2x

- Idx =

f

V2x + 1 - V2x -

1

= - I n I cosx | + C

X

(2x + 1)2 1 d(2x + 1) = ^^"^ ^ l)>/2x+_l

2

^ ^

3

^
+ C

3
Vay

F(x)

=

tan

X

+

dx

V(2x + 1) + V2x - 1

= - In cos x| + - [(2x + l)V2x + 1 - (2x - l)V2x - 1] + C .
' 6
6 0 I T i m ho n g u y e n h a m c u a ftx) =

—
cos x s i n

x

Cao ddng

Giao

thong

Van tdi -

1999
25

www.matheducare.com