Cẩm nang luyện thi đại học nguyên hàm tích phân số phức NXB đại học quốc gia 2012

C-UON

TRAN BA HA

AM NANG
LUYI^THI

DAI HOC

IIGUYEN HOM
*

*

THU VIENTINHBINHTHUAfO

mk nXr ikn i « i nc wfc cu M Mi


Cty TNHH MTV D VVH Khang Vi^t

LOI N O I £>AU

Phanli
I

N G U Y E N H A M - T I C H P H A N V A \SNG

Chuong

1:

r. .•,,

NGUYEN HAM

Tap sdch nay gom 2 phan:

A. T O M T A T LY THUYET

Phan I: Nguyen ham-Tich phan vaung dung

1. D i n h nghia: Cho ham soy = f(x) lien tuc tren khoang D

D U N G

^,
'

F(x) la nguyen ham cua f(x) tren D khi va chi khi: F '(x) = f(x) Vx e D

Phan 11: So phuc
Moi phan diroc trinh bay theo tung chirong, moi chuong bao gom cac
chuyen de, moi chuyen de du-oc phan thanh cac van de co ban, moi van de
bao gom: Tom t5t kien thurc - phuong phap giai - bai tap ap dung - bai tap
tu luyen. Cuoi moi chvrong deu co phan Bai tap tong hop va Bai tap luyen
thi bao gom cac bai tap nang cao duoc tuyen chpn qua cac de thi dai hoc va
cac de thi hoc sinh gioi.
Hi vong rang tap sach nay co the giup ich cho hoc sinh trong cac ki thi
hoc sinh gioi, ki thi dai hoc. Rat mong s u gop y cua doc gia va dong nghiep
de Ian xua't ban sau tot hon.

2. Tinh chat co ban:
+ Neu F(x) la mot nguyen ham cua f(x) tren D thi F(x) + C cixng la nguyen
ham ciia f(x) tren D (C la hang so)
+ Neu F(x) va G(x) la cac nguyen ham cua ham so f(x) tren D thi ton tai hSng
soCdeG(x) = F(x) + C
+ Ky hieu: jf(x)dx = F(x) + C ( l a h o nguyen ham ciia ham so f(x))
+ Neu f(x) va g(x) co nguyen ham tren D thi:
l[f(x) + g(x)dx = Jf(x)dx + l(x)dx
jkf(x)dx = kjf(x)dx, ke R

Trdn Bd Ha
Gido vien THPT Chuyen Le Quy Don - Dd Ndng
Tu nghiep tgi: lustitut de Recherche
Pour L 'enseignement des Mathe 'matiques
Paris-France

Nha sach Khang Viet xin tran trong giai thieu tai Quy doc gia va xin
long nghe moi y kien dong gop, decuon sdch ngay cang hay han, botch han.
Thuxinguive:
Cty T N H H Mpt Thanh Vien - Dich vu Van hoa Khang Vi?t.
71, Dinh Tien Hoang, P. Dakao, Quan 1, TP. H C M
Tel: (08) 39115694 - 39111969 - 39111968 - 39105797 - Fax: (08) 39110880
Hoac Email:

+ Neu Jf(x)dx = F(x) + C thi Jf(ax + b)dx = - F(ax + b) + C
a
+ Moi ham so lien tuc tren D deu co nguyen ham trenD.

Y<.'

3. Bang cong thiic nguyen ham ca ban:
C
ldx = x + C
Jx«dx=

fw^
je'""dx=

a +1

+ C a^-1)

J—
dx = ln x + C.
f

Jsinkxdx =

1

coskx + C

r
1
Jcoskxdx = — sinkx + C
k

^
+C

n ;

f J
a"
^
Ja''dx=
+C
Ina
J — ^ d x = tanx+C
COS"x

J — r - d x = -cotx + C
sin" X

•,. m


C 'dm nang luy^n thi DH - Nguyen ham - Tich phdn - So phm

I/uui

Ljy li\tin Ml V uv vn t^^nungnci

Bd Ha

sinx
F'(x)= — 1

B. C A C D A N G T O A N C O BAN:

KHAI NIEM NGUYEN HAM

Chuyen del:

Van de 1: C H l / N G M I N H F(x) LA MQT N G U Y E N H A M C U A f(x) T R E N D
Phuang phap: Chung minh F '(x) = f(x), Vx e D

1

s i n x ( l - c o s " x)
=

X

cos X

sin^ x
= — r -= t(x)

cos X

cos X

Vay F(x) la mpt nguyen ham ciia f(x).

•

2+x
Bai 5: Chung minh F(x) = x In

B a i l : Chung minh: F(x) = ln(x + V x " + 1 ) la mot nguyen ham cua:
f(x) =

cos

sinx

+ 21n(4 - x^) la mot nguyen ham cua ham so

f ( x ) = l n ^ ^ tren(-2;2)
2-x

tren R.

v-^

Giai

Giai

, 2+x
4x
4x
, 2+x
, ^
, ^
Taco: F'(x) = i n +5 - - - ^= ln-t(x) ,Vx6(-2;2)

D = R v i X + V x - + 1 > 0, Vx 6 R

2-x

1+= - 7 = J = = f(x)

x +Vx'+l

2-x

asinx + bcosx

Vx e R

Vx^+1

Bai 6: Chung mmh ham so f(x) =

Giai

f(x) = xcosx tren R.

ccosx - d s i n x

Giai

Ta c6: F '(x) = A + B

F '(x) = sinx + xcosx - sinx = xcosx Vx e R.

Bai 3: Chung minh F(x) = — j = la mpt nguyen ham ciia ham so' f(x) =

Vx

1

— T =

xVx

tren (0; oo)

Giii

V

=

-2

'

n

4

x^ =

J

1

cV;^

= f(x),Vxe(0;«>)

-2

1

Vay F(x) = —j= la mpt nguyen ham ciia f(x) = — j = r

Vx
Bai 4: Chung minh ham so F(x) =

1
3cos'x

xVx
1

cosx

la mpt nguyen ham ciia

ham so: f(x) = ^ ' " . ^ tren mien D = R \ - + kn; k e Z)
cos X
2

Vx 7t — + kn, ta c6:
2

(Ac - Bd)sinx + ( A d + Bc)cosx
=^
-^
^

c s m x + dcosx

Vay: F(x) = xsinx + cosx la mot nguyen ham cua f(x) = xcosx

'

.
> 0) co nguyen ham

dang: F(x) = Ax + Bin I csinx + dcosx | + C

Bai 2: Chung minh F{x) = xsinx + cosx la mpt nguyen ham ciia:

x'^

(c^ +
c s m x + dcosx

Vay F(x) = ln(x +Vx^ + 1 ) la mot nguyen ham cua f(x) tren R.

(

4x-

Do do: F(x) la mot nguyen ham ciia f(x)

F '(X) =

F'(x)=-2

4-x

csmx + dcosx

F(x) la nguyen ham cua f(x) <=> F '(x) = f(x), Vx
o (Ac - Bd)sinx + (Ad + Bc)cosx = asinx + bcosx, Vx
Ac-Bd = a
^
,
ac + bd „
bc-ad
Giai he ta co: A = — — - y ; B = — —
A d + Bc = b
c-+d^
c^+d'

{

Vay ho nguyen ham ciia f(x) la:
ac + bd
be - ad ,
^
^
F(x) =
r- + —

r- In csinx + dcosx + C
^'
c'+dc-+d'
Bai 7:
a) Tim a, b, c sao cho ham so: F(x) = (ax^ + bx + c) V2x - 3
. ^.
^ 20x--30x + 7 ^ . 3
cua ham so: f(x) = .
,
tren ( - , « )
V2X-3
2
b) Tim nguyen ham G(x) cua f(x) thoa man man G(2) = 0

la mpt nguyen ham

Giii
ax^ + bx + c
a) Ta co: F '(x) = (2ax + b) V 2 x - 3 +
,
V2x-3
(2ax + b)(2x - 3) + ax- + bx + c
V2X-3

Sax' + (3b - 6a)x + c - 3b
~

V2X-3
5

nil

ihiiiy, liivci]

- Xyjiv'ii

lu'iiii - Tirh

f)ficiii - Soptiijc - Trcin Bci Ha

Giai

F(x) la nguyen ham ciia f(x) <=> F '(x) = f(x), Vx e (—; oo)

xe^-Ce"-!)

5a = 20

Khi

x^

0 thi F

Khi

X =

0 thi F '(x) = I ' m

(x-De^+l

Xx)

F(x)-F(0)

3b - 6a = - 3 0 ^ a = 4, b = -2, c = 1.

,.
= Iim

-1

c-3b = 7
b) G(x) la nguyen ham cua f(x) nen: G(x) = (4x2 _ 2x + l ) V 2 x - 3 + C
G(2) = 0

..

C = -13. Vay G(x) = (4x2 _ 2x + l ) V 2 x - 3 - 13

Ta c6: F '(x) = (2ax + b)e'^ + (ax^ + bx + c)e'' = (ax^ + (2a + b)x + b + c ) ^

e"-!
*o

X

x->0

Bai 8: Tim a, b, c de F(x) = (ax^ + bx + c)e^ la mot nguyen ham cua f(x) = x V
Giai

e^-l-x

(x-l)e''+l

,-

2x

e'_l
2

2

khi x ^ O
= f(x) r:>dpcm.

Vay: F "(x) =
khi

X =

'V " '

0

De F(x) la nguyen ham ciia f(x), Vx e R thi phai c6

Bai 11: Chung minh Vx ^ 1 thi Vn e N', ta c6:

a =l
F '(X) = f(x) Vx € R r:> <^ 2a + b = 0 - > b = - 2
b +c=0

n.x""'-(n + l)x°+l

^ c =2

= 1 + 2x + 3x^ + ... + n.x-^'

(x-1)'
Giai

Vay: F(x) = (x^ - 2x + 2)e«
Bai 9: Tinh dao ham ciia ham so': F(x) = (x^ - 1) In 1 + x - x^In

Xet ham so' f(x) = 1 + 2x + Sx^ + ... + n x " '
1-x"

Tu do suy ra nguyen ham ciia ham so': f(x) = xln

Ta c6: F '(x) = 2xln 11 + x
,

, v6i X 9t 0 va X 7t 1

Ta c6: f(x) c6 nguyen ham la F(x) = x + x^ + x ' +... + x" = x

Giai

Theo dinh nghia ta c6: F '(x) = f(x), Vx ^ 1

-(2xln|x| + f r )

1-x"
« f(x) = x.1-x

x+1

n.x"-'-(n + l ) x " + l

1-x

> dpcm.

(x-1)'

= 2xln(l + x) - 2xln I x | + (x - 1) - x
Vay F '(x) = xln

1+x

Bai 12: Cho f(x) = xln - va g(x) = x^ln 4
4

-1.

Xet G(x) = F(x) + X => G '(x) = F '(x) + 1 = f(x) nen G(x) la nguyen ham ciia f(x;

Vay: G(x) = (x^ - l ) l n I x + x | - x^ln I x I + x la mot nguyen ham ciia f(x).
Bai 10: Chung minh F(x) =

khix;t0
khi

X =

Chung minh: f(x) = g'(x) - ^ x tir do suy ra: Jx.ln ^dx
Giai
g(x) = x^tn - => g"(x) = 2xln - + x^. - = 2xln - + x
4
4
x
4

0

( x - l ) e ^ +1

khi x ^ 0

Do do: - g'(x) - - x = xln - = f(x).
2 ^ 2
4

khi x = 0

f
X

1
X~
1
X
Suy ra: fx.ln —dx = — g(x) - — + C = — x^ln
J
4
2
4
2
4

la nguyen ham ciia f(x) =

X

4

+C


K^uiii

iiuii^ luy^n ini uii - ivguyeri nam - i icnpmm

- AOpmtC - Iran tsa na

Vamde2:TIM HQ NGUYEN HAM CUA HAM SO y = f(x) BANG DINH NGHIA
Phuong phap: Phan rich f(x) thanh tong (hieu) cua cac ham so' ca ban c6 the

^>

2

2x + l

tim nguyen ham bang each ap dung bang cong thuc nguyen ham ca ban, ap

4x'+4x--l

dung rinh cha't cua nguyen ham de tinh hoac dua ve dang nguyen ham cua

1
1
2x^ x" 1
1
f(2x-+x---—^dx=—+^-TX--lr|2x+l|+C
2
2
2 2(2x-l)

dx=

2x+l

ham so'hgp.

dx
dx

r

Bai 1: Tim hp nguyen ham cua f(x) = cosxcos3x

2(2x-l)

fV3^-V3x-2

Giai

S jx-dx - - J(3x -1)-' d(3x - 2) = - X V 3 ^ - -(3x - 2)V3x - 2 + C

f(x) = ^ [cos4x + cos2x]

3

Jf(x)dx = — J(cos4x + cos2x)dx = — [ — sin4x + — sin2x] + C
2
2 4
2
Bai 2: Tun hp nguyen ham aia f(x) =

2'"'

-5

x-l

-dx =

f2.2"

x-1

-
10^

10"

5.10^

1
= 2 j 5 - ' ' d x - ^ J2^dx =
5 J
5.2" ln5

x +1
Giii

9

+ C
5Mn5

Bai 5: Tim ho nguyen ham ciia cac ham so:
f(x)=l-

x +1

Jf(x)dx = J ( l -

x+1

)dx = x-21n|x + l

a) (sinx + cosx)^

b) sin-'x + cos^x

c) sin''2x + cos*2x

d) cos'^x

+C

Giai
a) J(sin X + cos x ) - dx =

Bai 3: Tim ho nguyen ham ciia £(x) = xe"

(1 + sin 2x)dx = x -

cos2x + C

Giai
b)

xe dx = — x e ' ' ' d ( x ' ) = - e ' ' ' + C
2J

=
4x'+4x^-1

1

V

b)

vxj

2x + l
2"*' — 5""'

d)

10^

Giai
1

8

/3
1
—+ —cos 4x dx = — X + — s i n 4 x + C
J 4
4
4

16

c) Ta c6: sin'>2x + cos"2x = (sin^2x + cos22x)(sin^2x + cos''2x - sin22xcos22x)
= l((sin22x + cos22x)2- 3sin22xcos22x) = 1 - - sin24x
4
'l-cos8x'

.1-2
4

-1

x + — + 2.x2.x3 dx

12 '
— + 3.x' + — x * + C =
2
7
2

dx

2

Bai 4: Tim ho nguyen ham ciia cac ham so':
a)

1 -cos4x

(sin^ X + cos'' x ) " d x =

+—xVx+C

5 3
_
= - + -cos8x
8 8

J(sin''2x + cos''2x)lx =

3
5 3
„ \
- + - c o s 8 x dx = - x + — s i n 8 x + C
r
8
64
8 8

d) Jcos' xdx = jcos'' x.cosxdx = J(l - s i n ' x ) - d ( s i n x )
9
1
= J(l - 2 s i n ^ X + s i n ' x)d(sin x ) = sin x - j s i n ' x + j s i n ' x + C


Cty TNHH MTV DVVH Khang Viet
r3x + l

Bai 6: T i m hp nguyen ham ciia cac ham so':
a) cosx.cos2x.sin4x

.

-7

r

-d(x + l) +

b) cos'x.sinSx
Giai

3
(x + l)-

a) Ta c6: cosxcos2xsin4x = ^ [cos3x + cosx]sin4x

Ai»
Bai 8: Tinh I =

= — [sin4x cos3x + sin4x.cosx] = - [sin7x + sinx + sinSx + sin3x]
Do do: fees X cos 2x sin 4xdx = — f(sin 7x + sin 5x + sin 3x + sin x)dx
J
4
4 J
' —cos7x+—cos5x+ —cos3x+
4^ 7
5
3
b) Ta c6: sinSx.cos-'x = sinSx .cosx

COSX

+C

(X

+1)

- d ( x + l)

+C

x+1

dx
sin" xcos'' X
Giai

Taco:

—
sin xcos X

, /

1 + cos2x^
3M

= ^ [sinSx.cosx + sin8xcosx.cos2x] = ^ sinSxcosx + ^ sin8x(cos3x + cosx)

64
''sin" 2x + cos^ 2x^
cot^2x^
= 64
= 64
.
sin"2x
, sin 2x sin 2x
sin 2x

1

cot" 2x

cot" 2x

cot^ 2x

sin 2x

sin 2x

sin 2x

s n r 2x

2

4cot-2x

2cot^2x

sin"2x

sin"2x

sin"2x

Do do: I = 32Icot2x + - cot^2x + - cot^2x] + C
3
5

= — (sin9x + sin7x) + — sin8xcos3x + — sinSx.cosx
4
4
4

71

Bai 9: Tinh

cos 2x + i^ .cos 3x + - dx
3J
4

= ^ (sin9x + sin7x) + ^ (sin! 1 x + sin5x) + ^ (sin9x + sin7x)

Giai
1

3
3
1
= —sinllx + — sin9x + — sin7x + — sm5x
8
8
8
8
Do do:
1
sinSx.cos^ xdx =

2
cosl Ix

88

3
cos9x

24

cos7x

cos5x + C

56
A

49

A

B
A + B(x + 1) Bx + A + B
- +
^=
'
=
^
(x + 1)'
(x + 1)(x + 1)'
(x + 1)'
Dong nha't ta c6: B = 3, A = -2
10

f - 4 x ' + 9x + l

B

9-4x"

cos 5x + —

I2J

+ cos

X

UJ

n
+ —sin x 2
12

dx

^=
^+
(x + 1)'
(x + 1)' (x + 1)'

3x + 1
b) Suy ra ho nguyen ham ctia ham s6'f(x) = —'r(x + 1)
Giai

2

f(x)dx = — s i n 5x +
10
12

BailO: Tinh I =

a) Xac djnh cac he so' A, B sao cho:

Taco:

Do do:

1

Bai 7:
3x +1

1

Ta c6: cos 2x + — cos 3x + 4;
3

Giai
1=

4x'-9x'-l
4x"-9
xdx

dx =

1
—
6-'Ux-3

x -

1
( 2 x - 3 ) ( 2 x + 3)J

2x + 3 j

dx

^

2
2

dx

2x-3
^In
+ c
12
2x + 3
12

§ai_ll: Cho cac da thiic: fn(x) = 1 + 2x + 3x- + ... + n.x" '
va gn(x) = 1.2 + 2.3x + 3.4x2 +

+

+ ] )x»-"
11


CtyTNHH
a) T i m nguyen ham F(x) cua fn(x) thoa man man F(0) = 1. Suy ra bieu thuc t h u gon

MTV D VVH Khang ViC't

y | n de 3: T I M M Q T N G U Y E N H A M C U A H A M S O f(x) T H O A M A N D I E U

cua fn(x)

KIEN CHO T R U O C

b) C h u n g m i n h : gn(x) = f 'n*i(x). T i m bieu thuc t h u gon ciia gr.(x).
Giai
a) fn(x) = 1 + 2x + 3x2 +

+n

F(x) = x + x^ + x^ + ... + x" + C

X"-'

F(0) = 1 o F(x) = 1 + x + x2 + ... + x" = 1

x"^'-l

khi

x-1

Vay F(x) =

n +1

khi

X

X =

(x-1)^
n(n + l)

b) fn+i(x) = 1 + 2.x + 3x2 +

x-1

(tong cua cap so'nhan)

+

A p d u n g p h u o n g phap t i m ho nguyen ham de t i m F(x) + C

+

D u a vao dieu k i ^ n cho t r u o c de xac d j n h hang so'C.

Bai 1: T i m n g u y e n h a m F(x) cua h a m so'f(x) = cot^x biet F( — ) = 0
4
Giai

^1

f c o t ^ x d x = f ( l + c o t ^ x - l ) d x = f ( — \x + C
J

sin x

1

nx"^' - ( n + l)x" -1
F ' ( x ) = f„(x) =

1-x

p h u o n g phap:

khi

X ^

1

khi

X =

1

+ n.2"-' + (n + l ) x " suy ra:

f •n.i(x) = 1.2 + 2.3x + 3.4x2 + _ + n(n + l ) x " - ' = g„(x)

F(x) = - c o t x - x + C ; F ( - ) = 0 < » - l - - + C = 0
4
4

Hay

C =1 +

; Vay: F(x) = -cotx - x + 1 + ^

Bai 2: C h o f(x) = sin^x(1 + tanx) + c o s \ ( l + cotx)
T i m n g u y e n h a m F(x) cua f(x) biet F( — ) = 1
4
Giai
Rut gon f(x) ta c6: f(x) = sinx + cosx
Jf(x)dx = sinx - cosx + C => F(x) = sinx - cosx + C
F(-) =l < : ^ s i n - - c o s - + C =1 « C = 1
4
4
4
Vay F(x) = sinx - cosx + 1
Bai 3: C h o f(x) =

?

. T i m nguyen h a m F(x) biet F( - ) = 0

l + cos2x

3
Giai

f(x) =

l + cos2x

Jf(x)dx=-

2

F(^) =

3

=

f — ^ d x =
cos- x

0<^1.V3.C =
2

—

2cos X

- tanx + C => F(x) = - t a n x + C

2

2

0 ^ C = - ^

2

V§yF(x)=itanx

12

13


Cam

iwn}^

luy('n

thi DH

Bai 4: C h o f(x) =

•1.

- Nguyen

ham - Ticli plum

- So phi'rc

- Trdn

Bd

^ ^, F(x) la m o t nguyc"n ham cua f(x) thoa m a n m a n : F(2) =

••-

•

T i n h F(5)

.

••v.i'

f(x) = — ^
X

- !

F(2) = I

CO

F(x) = in
C =0

I

X -

Ux) = In

r = — - I n — = — + lnN/2
2
2
2

• • r^!.^

y = F(x) d i qua d i e m M ( —; 0)
6

ij

Giai

Bai 5: C h o f(x) = vcos^ x + 4 s i n " x . T i m nguyOn h a m F(x) cua f(x) bie't:
71

I

4

4

f(x) = ^cos^ X + 4(1 - c o s ' x ) = 2 - cos^x =

f(x) =

1
— ^ F(x) = - c o t x + C

sin' X

(3 - cosx2x)

<=> - c o t - + C = 0c=>C = c o t - =
6
6

X

^
1
F{-)=
+C = — c^C
4
8
4
4
1
1
D o d o F(x) = — (3x - — sin2x) -

•

Vay F(x) = - c o t x + ^/3

F(x)= ^ ( 3 x - ^ s i n 2 x ) + C
1

••

D o t h i y = F(x) d i qua M ( - ; 0 ) « F( - ) = 0
6
6

Giai

371

,. *

Bai 7' C h o f(x) = — \ — . T i m nguyen ham F(x) ciia f(x) bie't d o thj h a m so:
--—
sin X

Do d 6 F(5) = In4 = 21n2

71

Viet

...

1I +C

iX-

M r\l)\I Khang

TNHH

D o d o F(x) = tanx In(sinx) + ^ Xnyjl

'r>'> f>Tii^t^f'-^

Giai

Cty

Ha

^371
=
8
371
—

Bai 6:
a) C h i i n g m i n h F(x) = tanx In(sinx) - x la mot n g u y e n h a m cua:

-1

Bai 8: C h o bie't F(x) =

la nguyen ham cua f(x). T u n f(x - 1)
x +1
Giai

f(x) = F ( x ) =

^x-lV

2

,x + 1

( x + 1)-^

2
Dodo: f ( x - l ) = — r
X"

'7

f(x) = (1 + tan^x) In(sinx) tren (0; ^ )

b) T i m n g u y e n h a m F(x) cua f(x) bie't F( —) = —
4
4
Giai
a) V x e (0; - ) , F ( x ) = — ^ l n ( s i n x ) - ^ ^ ^ t a n X - 1
2
COS" X
sin x
F '(x) = (1 + tan^x) In(sinx) = f(x)
Vay F(x) la m o t n g u y e n ham cua f(x)
b F 4
14

= - o t a n - In(sin-)- - +C= 4

4
4
4
4
15


Cdm nang luyeti ihi DH - Nguyen ham - Tich phciii - So phi'rc - Trdn Ba

PHl/ONG PHAP T I M NGUYEN H A M

Chuyendel:
Van

Cly TNHH AfTV D VVH Khang Vi(H

Ha

de 1 : T I M H Q N G U Y E N H A M B A N G

3 1 [71

P H U O N G PHAP D O I BIEN SO

Phuong phap: Co the doi bien so theo hai each sau:

.

^ s /(sin X -

c o s x)-

gai_2: Tim ho nguyen ham ciia ham so'sau: f(x)=

a) Dat u = cp(x) la ham so'co dao ham thi:
Jf(x)dx = |f(u)du =F(u) + C

Giai

b) Neu f(x) Hen tuc c6 dao ham, dat x = (p(t) th'i: Jf(x)dx = Jf{(p(t))(p'(t)dt

Dat X = atant => dx =

B a i l : Tim ho nguyen ham cua cac ham scYsau:
b) f(x) =

c)f(x) =

d) f ( x ) =

Vx^-Vx

dx
x"'-4

yjix'+a')'

sinx + (COSX
.
vsin X - cosx

2
a" •'

+C

20
dx

20 U - 2

x' + 2

dx

= \

U +

2J

du =

—In

20

u-2
u +2

+C

a)

b)

.5

j f (x)dx = 6 j ^ d . =6|il^ = 6 j ( . . I . ^ ) d .
—+t + lnt-l + C = 6
2

16

.

du = (cosx + sinx)dx
fdu

a' J V l + tan-t

1
X
costdt = — s i n t -f C, vai t = arctan —
aa

t
u-^
„
lu^du = ^ + C

•'vu •
A

^^=jTr"

1

COS X

COS X

cosx

cos'x

l-sin'x

Vay

r.Datx =

Ta c6: dx = 6tMt, do do:

f sinx + cosx
hi•"Vsinx-cosx

dt

b)f(x)=^
sm X

Dat u = sinx =>

+ C

x'-x^

d) D|t u = sinx - cosx

^ a V ( l + tan't)-

dt =

Giai

du
x'-2

(1 + tan-1)

a)f(x)=—Lcosx

b) Dat u = x^ => du = S.x^dx

-In

t

Bai 3: Tim ho nguyen ham ciia cac ham so'

j x V ' x - - l d x = ^ jyf^du = - A / 7 + C = \^{\'-\f

11--4

a"

= a(l + tan2t)dt

cost

a) Dat u = x^ - 1 => du = 2xdx, do do:


= a

dt

Giai

x"'-4

^dt
CDS'

a) tXx) = x V x - - 1

4

+C = |Vl-sin2x +C

-u^

1 / 1
2 J 1+ u

1+u
du = —In
+ C
1- u
2
1-u

1 -i-sinx
= -ln
+ C
cosx
2
-sinx
X

1
— ^ — ( 1 -i-cot- x)dx
sin- X

Dat u = cotx => du =
+ C

f du

dx

dx
sin^

f (x)dx =

Do do:

f-T-T—

•"sin X

1
^—dx
sin-x

= -k^ + U')du - (u + — ) + C = -(cotx + - cot-^x) + C
3
3

Bii_4: Tim hp nguyen ham ciia ham so':
a) f(x) =

1
xlnxln(lnxj

b) f(x) =

Va" - x"

TH(/ Vf^N TINH BINH THUAW
17


Clyjmm

Ccim nang liiyen thi DH - Nguyen ham - Tich phdn - So phiic - Trdn Bd Hd
Giai
a) Dat u = In I In(lnx) |

MTV

D VVHKhang

VH

2^ 3^

du = i l l l i l E ^ j x
In(lnx)

Giai
^3^

du =

x l n x l n ( l n x ) - dx, do do:

•

''>i-'• •

'•••t<-

_dx . Dat
3

dx

^

=>du =

-I
du

J=

b) Dat X = asint-—
,u - 1

In

J=
fVa' - x" ,
f a cos t.a cos tdl
—dx=
:

J

Y

a

sin

rcos" t ,
^
dt

t

ff,
1 >
du
f(x)dx = a
= a 1 + -^
u'-l
\

a" -

X'

V

2(ln3-ln2)

3^-h2''

.
a)

fcos X ,
f(l - s i n - x )
< I
dx = \
^cosxdx =
sin X
sin X
V sin
1

1

b) f
'
dx = f (sin x + COS x ) "
• ' l + sin2x
J^'^i
'

, do do

sin

Va" -

X'

-a

dx

2 •' . '
sin -

A

1

Va" +

X"

+a

Bai8:

4j

Tinh j(x-l)e^"'-"''"'dx
Giai

Dat u = x2 - 2x + 3 => u' = 2(x - 1 )
(x - l ) e ' ' ' - ' " M x = i je-u'dx - i Je-du

Dat u = (1 + xe") => du = (x + l)e''dx, do do:

2

U=COt2xr:>u' = -

= -2(1

+cot22x)

sin" 2x
1

du = In
u

1 _x-"-2

2
Giai

.u -1

71
X+ —

x+

8419: Tinh: J(l + cot^Zx) e-'^-dx

u(u-l)

1
—cot
2

+ C

-dx
xO + xe")

rf

d(sin x ) :-tk

4J

x +1

du

X

X

sin^x + C

< " " ^ " -dx
xe^(l + xe^)

r

+ C

1 + sin2x

Giai

1=

3^-2'

-In

2
dx

vi cost = V l - s i n - t = j l - — =

+ ^ In

-i-C =

b)f(x) =

+ C

= a cost + — I n
2
cost + 1

X"

u-Hl

= In sinx |

cos t - 1

f (x)dx - V a ' -

2(ln3-ln2)

u-1

Giai

1,
du = a u + - l n
+ C
u +1
2

u+1

-In

a) f(x) = —
sinx

u-du

1
= a 1+ —
2 u-1
V

du

Bai 7: Tim ho nguyen ham cua cac ham so':

J sin t

Dat u = cost => du = -sintdt

/

u -^ 1^

2

V a ' - x" = aVcos" t = a(cot) = acost

1=

3
In—dx
2

"du = u + C = In I {In(lnx)) I + C

xlnx.ln(lnx)

Bai 5: Tinh I =

3

u =

u-1

+ C = In

xe^
1 + xe^

+ C

J(l + cot22x) e-"'2'
Je".u'dx =

e-^'^ + C

e

+c

;1j

+ C

(

iiiir.yi

/in cii llii

• ;/ - / A /; plum

/'// - A i v / i iv;

- So

phllV

OyTNHH

BaHa

V a n de 2: T I M H Q N G U Y E N H A M B A N G P H l J O N G P H A P N G U Y E N
HAM Tl/NG PHAN

Lai dat u = Inx => du =
dx

Phucmg phap: Gia six u(x), v(x) la cac ham so c6 dao ham lien tuc khi do ta c6:

Ju(x)v'(x)dx = u(x)v(x) hay

udv = uv -

v(x)u'(x)dx
Vay

vdu

Chu y: Cac dang sau:

+

P(x) .sin(ax)dx , P(x) cos(ax)dx: Dat u = P(x), dv = sinax (cosax) .dx

+

|P(x) e-^^dx: Dat u = P(x), dv = e-'^dx

+

e"" sin(bx)dx hoac je'" cos(bx)dx

X

/inx V ,
1,5
^
dx = — l n - x + 2 - — i n x - — + C
X
X /
I X ;

X

a) f(x) = x^lnx

b) f(x) =

|(x + l ) s i n 2 x d x =

cos2xdx

= - — (x + l)cos2x + — sin2x + C
^ l + cos2x^

dx

f Inx

dx

I = — J ( x + l ) " d x + - ( x + l)"cos2xdx
2
Ta c6:

, do do:
dv = xMx => v =

•(x + i ) M x = ^ ^ ^ + C

Xet J = f(x +1)" cos2xdx . Dat u = (x + 1)^ r:> du = 2(x + l)dx
•

dv = cos2xdx => V = — sin2x
2

fxMnxdx = — I n x - - f x ' d x = — i n x - — + C
b) Dat u = In^x => du = 21nx

+ l)cos2x + ^

b) 1= |(x + l ) - c o s - 2 x d x =J(x + l)-

Giai

a) Dat

b) f(x) = (x + ^fcos'x

v

Bai 1: Tim hp nguyen ham ciia cac ham so'sau:

20

X

dv = sin2xdx => v = - ^ cos2x

Jvdu don gian hon.

Do do: J = - (x + 1 )2sin2x + |(x + l)sin 2xdx

dx

Xet A :

1
v= —

(x + l ) s i n 2 x d x . Dat u = x + 1 =>du = dx
dv = sinZxdx => V = — cos2x
2

dx = - - l n ^ x + 2 t e d x
Y

X"

X

1

a) D|itu = x + 1 =:.du = dx

+ Tong quat: Phan tich f(x)dx thanh u va dv sao cho: t u dv suy ra duq^c v va

fflnx^

=>

Giai

Dat u = e-'"; dv = sin(bx)dx hoac dv = cos(bx)dx

dv =

inx ,
1,
—^dx = — I n x

1
=:>V=

a) f(x) = (X + l)sin2x

P(x) Inxdx: Dat u = Inx, dv = P(x)dx

dx

—
X"

dx

Bai 2: Tim ho nguyen ham ciia cac ham so:

+

u = In X => du =

dv=

MTV DVVH Khang Vu

J

X"

A = - - i ( x + l ) c o s 2 x + - fcos2xdx = - - ( x + l)cos2x + ^sin2x + C
2
2 J
2
4


Cam

nang

luyen

thi DH - Nfjiiyen

Vay |f(x)dx =

6

2

ham - Tich phiin

- Sd phirv

- Trdn

^ ( x +1)^ sin2x - ^ ( x + l ) c o s 2 x + l s i n 2 x + C

^ O i : ^ 2 L + I ( x + i)2 s i n 2 x - i ( x + l)cos2x + i s i n 2 x + C
6
4
4
8
Bai 3: T i m h o n g u y e n h a m cua cac h a m so'sau:
a) f(x) = (x2 + 1 )e2-dx

C . / y IWlltl

Bd Ha

XetJ=

j e " C O S x d x . Dat

sin" X

u =x
a) D a t
+1

c-'^ - Jxc-^^dx

Vay: | ( x ' + l ) e ^ " d x =

x +1

C

e

+ - C

+C

cos" X

= tan X

f d ( c o s x ) = xtanx + In | cosx I + C
cosx

a) f(x) = e"2^cos3x

b) f(x) = sin(lnx)
Giai

d u = -2e-2^dx

- ifs.:c

e~"'cos3xdx = — e - \ s i n 3 x +
J

Vay: Je^'''dx = 2 ( V x - l ) e ' ' ^ + C
sin xdx va J = Je" cos xdx
Giai

dv = sin xdx => v = -cos xdx

V

Bai 6: T i m ho n g u y e n h a m cua cac h a m so':

Dod6:I=

D o d o : Jte'dl = te' - je'dt = ( I - l)c' + C = (Vx - Oc"^ + C

22

dx

1 > ' ?'/ ^ /K^*

d v = cosSxdx => V = — sin3x

du = dt

> du = c ' d x

-V= -cotx

=> du = dx

a) Dat u = e-2>=

d v = e'dt => V = e'

u =e

sin^ X

B= xtanx +

J e ^ d x = 2Je'tdt = 2jte'dt

Dat

dv =

A

b) D a t t = V x o t2 = X => 2tdt = d x

Bai 4: T i n h I =

dx

u =X
b) Dat

= - ( 2 x 2 - 2 x + 3)e2- + C

Dat u = t

=>du=dx

A = - x c o t x + In I sinx I + C

d v = e^'dx => V = — e^^
— e dx = —e
2J
T

dv =

.< :. - ci

., .
rd(sin
•d(sin X
x)
Do d o : A = - Xcot x +
J ssin
in X
X

Xet J = Jxe-''dx. D a t u = x=>du = dx

= —e
2

,

Giai

1

^

x
^x
cos" x

Bai 5: T i n h A = f - ^ - va B =

— e^*

I = J(x- + De^^dx =

(2)

e"
I-Do do: I = — (sinx - cosx) + C va J = — (sinx + cosx) + C

a) D a t u = x^ + 1 => du = 2xdx

X

=> du = e"dx

e"" sin x d x = e^sinx - 1

J = e'sinx -

r tcr

dv = cos xdx => V = sin xdx

b) f(x) = e''^

Giai

dv = e^'dx r:> V =

u = e"

mi V UV Vll l\.nun^

XetJ=

3

f e - ' s i n 3 x t l x e D a t u=e

c'-'sin3xdx
3

'-ryiv^;.

d u = - 2 e ^^dx

d v = sinSxdx => v = - ~ c o s S x

3
J = — e^^VosSx - — e " ' c o s B x d x
3
3

. do do I = -o^cosx + j e " cos xdx (1)
23


C 'ty TNHH MTV D VVH Khang Vii^t
1
2
1
D o d o I = — e-^'sinSx + — (
e-2^cos3x
3
3
3
1
2
4
1 = — e"^'e^^cosSx
1
3
9
9
<^

2

3

D o do: I = f V x " + k d x = x-y/x" + k -

I)

= XV

= - ^ ( 3 s i n 3 x + c o s 3 x ) e " - ' => I =--p;-(3sin3x+cos3x)e"^'' + C
u = sin(lnx)

b) Dat

=>du = —cos(Inx)
X

dv = dx

=o V = X

A = Jsin(lnx)dx = x s i n ( l n x ) -

XetB =

, ^,
, cio d o :

cos(lnx)dx. Dat

dx

dx
- f ^ ^ T ^ ^ d x = XV ^ ^ k - 1 + k f - ^
Vx- + k
•'Vx' +

21 = x V x " + k + k i n X + V x ' + k
+ C

I = -!-f x V x " + k + k Injx + V x " + k
2V
'
Bai 8: Ti'nh Jxsinxdx

Giai

cos(lnx)dx

1 .
u = cos(ln x) => d u = — sin(ln x)dx
X

d v = dx
B = xcos(lnx)+

^

Vx-' + k

Dat u = X => d u = dx

d v = sinxdx => v = -cosx

j. .

Do do: Jxsinxdx = -xcosx + Jcosxdx = -xcosx + sinx + C

V = X

Bai 9: Ti'nh J(x - 1)e^dx

sin(lnx)dx

Giai

D o d o : A = xsin(lnx) - xcos(lnx) - A

Dat: u = x - l = > d u = dx
d v = e^dx => V = e"

Vay A = ~" (xsin(lnx) - cos(lnx)) + C

Do do: J(x - 1 )e>^dx = (x - 1
Bai 7: T i m ho n g u y e n ham ciia cac ham so:
a)f(x)=^'"(^^^^>
+ x^

- je^dx = (x - 1

-

+ C = (x - 2)6" + C

Bai 10: T i n h j x l n x d x
Giai

b)f(x)= Vx- + k
Dat u = Inx => d u = —
Giai

a ) D a t u = ln(x + V l + x- ) = ^ d u =
dv

1=

fxln(x + V l + X - ) ,
n
7,
r
7
,
^ x = Vl + x M n X + Vi + X- Vi + - '

u = Vx" + k
dv = dx

24

Do d o :

dx=> V = V l + x"

= V l + x" In X + V l + X

b) D a t

d v = xdx => V =

dx

-x +C

=x>du=

^
dx
Vx-'+k

V = X

r.
dx

X in xdx — i n x - •-dx = — I n x - — + C

2

2

2

4

S a i l l : T i n h 1x^(2 - Sx^fdx
Giai
D a t t = ( 2 - 3 x 2 ) = > d t = -6xdx
JxX2 - 3x2)8dx = y
f2-t
3

s
.t' {
.

6,

(2 - 3x2)«xdx

dt = l j ( l " - 2 t ' ' ) d t
18 J

1
.t"'-J_t%C = —(2-3x^)'"--(2-3x^)"+C
180
81
180
81
25


Ccim ncing hiyen ihi DH - N}^iiycn ham - Tich plidn - So phirc

Chuyen de3: N G U Y E N H A M CUA
Van

Trdn Ba Ha

CIV TNIIIIMTVDVVHKhang

C A C H A M S O CCf B A N

de 1: N G U Y E N H A M C U A C A C H A M SO

H U U

gai 2: T i ' " ho nguyen ham cua cac ham so'
^
a) f(x) = 3

TI

Phuang phap: De t'lm ho nguyen ham ciia cac ham so'dang

P(x)

,

i ;

3x- + 1
_ .—
7

-2x- -5x + 6
Giai

3x' + l
^^^x- ~5\ 6

5.^ I T ?

+ Ne'u bac ciia P(x) nho han bac cua Q(x) th'i phan h'ch Q(x) thanh tich cac

3x' + l

'

thuc don gian (mau so' la cac thiia so' bac nhat, bac hai 6 tren) de tim ho
nguyen ham.

~ T *

„
14
x = 3=>C= —

1

dx
(x + a ) '

1

l - k ' ( x + a )k-l
'

'>

+c

+ Chu y phirong phap dong nhat da thiic khi phan tich
b)

Bai 1: Tim hp nguyen ham cua
1

u\/ \x + 3
X" + 2 x - 3

a)f(x)=x'-3x + 2

Dodo: f — ^ — ^ '
dx = - - l n x - l + — l n x + 2 + — l n x - 3 + C
•"x - 2 x - - 5 x + 6
3
15
5
3x + l
A ^
B
^ C _ A(x-l)(x-2) + B(x-2) + C(x-l)(x-l)-(x-2)

B

(A + B ) x - 2 A - B

x-2

(x-l)(x-2)

A
(x-l)(x-2)

Dong nhat ta co:

r-dx

X--3X+2

x-I

4x + 3

4x + 3

x-+2x-3

( x - l ) ( x + 3)
A +B=4
3A-B = 3

- +

A

f dx
J

X -

2

= ln

x-2

Vay

3x + l
(x-l)-(x-2)

dx = - 7 1 n x - l +
+ 71n

x-l

B =l

-2A-B =I

(x-l)-

x-2

x-l

(x-ir(x-2)

x = - l => A = - 7

A--1

[A + B - O

dx

Dong nha't ta c6:

x-I

+ -

x-l

K h i x = l = 5 B = - 4 ; x = 2=>C = 7

Giai

4
x-l

x-2
x-l

+ 71nx-2 +C

+C

M i 3 : Tim ho nguyen ham ciia cac ham so:
+ C

7

a)f(x) =

B

4
B= ^
4

X"

(x-l)'

x'-7x-+14x-8
Giai

X - l+ x + 3

r 4x-3
,
7 r dx
9 r dx

7,
. 9 ,
\
dx = - 4 • ' x -+l - 4 J x + 3= - l 4n x - l + — l n x + 3 + C
•"x + 2 x - 3
4 J Y - I
4JY+'^
4
4
26

.

Dung he so bat djnh: x = l =>A = - y ; x = -2=:>B = |j

+ Chu y cac ho nguyen ham C O ban
r
dx
= ln X + a + C
'x + a

b)

>• •<• / ' •

(x-l)(x + 2)(x-3)

thuc de du'a ve truong hop tren.

Do do:

(x-l)^(x-2)

A
B
C
+
x - l ' x + 2 • x+--3

(x-i)(x + 2)(x-3)

+ Ne'u bac cua P(x) Ion hon hoac bang bac cua Q(x) thi dung phep chia da

X--3X + 2

3x + l

_ A(x + 2)(x - 3 ) + B(x - l)(x - 3 ) + C(x + 2)(x - 1 )

thira so bac nhat va bac hai roi phan tich thanh tong (hieu) ciia cac phan

a)

. . ,-. ,
b)l(x) =

, voi P(x),

Q(^)
Q(x) la cac da thiic ta thyc hien:

Viet

(X-l)'
r

x'-l +l

x +1

(x-l)'

(x-l)'

+ -

(x-l)'

(x-l)'

(x-l)'

(x-l)'

x^

J r — : 5 - d x = f(x-1)-'dx + 2 f ( x - l ) - ' d x + f(x-1)^dx
( x --l1)
•>-2
1 J „

- 6 x -J + 4 x - l ^
+ C
2(x-l)- + 3(x-l)^ 4 ( x - l ) ' + C =
I 2 ( x - 1 ) ':

27


C 7)' TNHH MTl nVVH Khang Viet

Cam nang luy^n thi DH - Nguyen ham - Tich phdii - So pht'rc - Trdn Bd Ha
b) Ta c6:

+2x + 6

X

x^-7x^+14x-8

+2x + 6

B

(x-l)(x-2)(x-4)

x-1

Giai

C

• +

•

x - 2 ' x - 4

Dat u = X + 1 => d u = dx

- A ( x - 2 ) ( x - 4 ) + B ( x - l ) ( x - 4) + C ( x - l ) ( x - 2)

dv =

(x-l)(x-2)(x-4)

dx
(2x-l)'

D o do: k =

D u n g he so bat d j n h (gia trj dac biet)
x = l r : > A = 3 ; x = 2 r : ^ B = - 7 ; x = 4=>]< = 5
f

= 31n

x"+2x + 6

-dx = 3
•7x- + l 4 x - 8

IX- 1 I-

x-i

-dx-7

dx
x-2

i x - 2 i + 51n I x -4 I + C = In
X -

71n

+ 5

r

.X

x-2

dx

-4

5

-+c

dx
x ' - g x ' + ie

A

A

(x-1)'

(x-l)'

( x - 2 ) . 2 . ( x + 2)-

(x-2)-

D

+

x-2

(x + 2)-

x+2

+•

Vay:

,

x' + l

^

Ml

33

-dx =
x ' - 8 x - + 16
2

+

i6(x-2)

— in \ - 2
32

Bai5:TinhJ= f
^
•'x'+6x-+5

Taco:

I

x'+ex'+S

xdx

(x'+3)-'-4

_ I r dt

1 ff

•' x' x' '' +' +6 6x x- +
- +5 5 " 22 - Jt
't'-4
Bai 6: Tinh k =

28

j

x + 1
(2x-l)'

dx

X - I

3

3

1

( x - I ) ^ • +( x - T ) ^ - +X•- l

dx

C =l
3

2(x-l)-

x - l

+ In X - 1 + C

3x + 7
' ' -dx
+4x + 3

3x + 7
x-+4x + 3

Suyra:

^ -+
o 3 x + 7 = (A + B)x + 3 A + B
x+ 1 x+3
A +B =3
3A + B - 7

=^ A = 2,B = 1.

^ 3x + 7
dx = 2 f ^ + p ^ = 21n|x
\1 + In x + 3 + C
Jx + l
•'x + 3
'
x" + 4 x + 3

Phuong phap:

Ho nguyen Mm dang: [R
R

D a t t = x2 + 3 =:> d t = 2xdx, d o d o :
f

r-+-

Yande2: N G U Y E N H A M M Q T S O H A M V 6 T I
Giai

I

C

(X-l)-

<=> Q _ 3, do d o : M =

3

D o n g nhat ta co:
129
31
16(x + 2)• + —
32 l n x + 2 + C

+

Giai
Ta co:

„
, .
33
_
-31
„
127 ^
129
Ta co: A = — ; C =
; B=
;D =
16
6
32
32

+C

•' X -

D o d o : Bx--' - 16x + 1 = A ( x + If + B(x - 2)(x + 2)^ + C(x - If + D(x - 2)2(x + 2)
Thay Ian l u o t cac gia t r j : x = 2, x = - 2 va x = 0, x = 1

4(2x-l)

A =3

Bai 8: T i n h N = f ,

C

2(2x-l)-'

D o n g nhat ta c6: x^ + x + 1 = Cx^ + (B - 2C)x + A - B + C

=^k= -

B

B

x- + X + 1

x ' ' - 8 x - + 1 6 ~ ' ^ ^ ( x - 2 ) - ( x + 2)^
8x^-16x + l

1

dx

(x-l)-

A-B+C=l

8x^-16x + 1

2-'(2x-l)-

X" + X + 1

B - 2C = 1
Giai

X +

dx

2(2x-l)'

C = I

x'+l

Bai 4: Ti'nh I =

Dat

2(2x-l)=

x+1

Bai 7: T i n h M =

x-4

-1

V =

8

I
t-2

x^+1
t-2
+C
d t = - In
+ C =-ln
t+2
8
t+ 2
8
x^+5

X,

rax + b^ ti

f a x + b^

,cx + d ^

V cx + d j

Goi k la boi so c h u n g nho nhat cua cac mau so:
n

dx

_
S

Dat t'^= ^ ^ ^ ^
cx + d
29


Cam nang hiyen /hi DH - Nguyen ham - Tich phcin - So phi'cc - Trdn Bd Ha
Oat 2x + 1 = t* => 2dx = 6tsdt <=> dx = 3f^dt

b) H p nguyen ham dang: J R X, Vax" + bx + c Jx
Neu a > 0: dat t + V a x = Vax" + bx + c

J

Jt

-1'

-1

t+l +

J

dt
t-U

Neu c > 0 : dat xt ± - / c = V a x ' + bx + c
= 3[Y+

Neu ax^ + bx + c c6 nghiem xi, X2, dat Vax' + bx + c = t(x - xi)

vrri+Vx+T
Giai

^
ax2

+ bx + c = a

A

X +—

Dat X + 1 = f' => dx = 6^dt, ta c6:

roi dat II = X + -— de dua ve cac dang:
2a

4a'

| R , ( U . \ / U ^ + a - ) d u : dat u = atant

xdx

g ^ : Tinh C = [

(Truong hop a < 0 hoac c < 0 thi dat x = — dO diia ve dang tren)
u
Chu y: Co the huu ti hoa bang each bie'n doi
b

t + l n l t - l l ] , v 6 i t = V2x + I

K t " - l)t'dt
J

t +t'

I

^ r(t" - l)t'dt
— (,

t+i

^ r(t'
=

+ Dt'dt

5

i+i

C = 6 (t- - t + l ) ( t * - t ^ ) d t = 6 ( t ' - t ^ + f - r + f - t ^ ) d t
or.

'R,{u,>/u--a-)dii:ctatu=

C= 6

cost

+ C, vol t = 'Vx + 1
9

8

7

6

5

4

J R , ( U , Va' - i r )du : cTat u = asint
Bai 4: Tinh D = i
c) H o nguyen ham dang:

Vx" +2x +

f
(Ax + B)dx
•'(x-a)"Vax'+bx + C

Giai
Dat t - x = V x - + 2 x + 3 =i> t2 + x 2 - 2 t x =

Dat X - a = t

•

•r.-n •••KV

+ 2x + 3

t=-3

^
1 t-+2t + 3
dt
=> dx = —
(t
+

l)=
.
2(t + l)
2
t'+2t + 3
t=+2t + 3
dt
D = 1 f (t + i)=
1 rJt+llL dt
2 JJ
t'-3
2 •'t-+2t+3
t2(t + l)
2(t + l)
= X

Bai 1: Tinh A =

f^^^Z^jx
•'x(Vx + l)
Giai

A = J - ^ ^ — p ^ d x . Dat X = t« ^ dx = 8tMt
x(x^+l)

= f

D o do ta co: A =

Kt=-I)8t'dt

=8

t-i

t+

= l n t + l + C = ln x + i + V x ' + 2x +3 + C

dt

dx

S i i S : Tinh E =
• tdt

Vx^ +6x + 8
= 4ln!t- + l!-8arctant + C

Giii

t- + l
dx

Bai 2: Tinh B =

^ ^ t t - x = V x ' + 6 x + 8=>x =

2(t + 3)

V(2x + 1)' -V2X + 1

t^+6t + 8
, dt

Giai
dx

B=
J

2

(2x + l ) ' - ( 2 x + l)=
30

t-8

dx=l:±^d.,dod6E=i|-(*^^)
2(1 + 3)-^

t-

t^-8

2(t + 3)
3i


Cdm nang luy^n thi DH - Nguyen ham - Tich plidii - So phi'rc - Trdn Bd Ha

E=

f - ^ = lnlt + 3 + C = lni x + 3 + -\/x- +6\ 8
Jt + 3
'

(lyTNHHMTV

phitt»ng phap
a. Dgng R(sinx, cosx).

Giai

dx=

X= -^^-^
2(t-2)

0-2)^+4

r-8

F=

X

= 1 - 2(t-2)

(t-2)-^

K(t-2)^+4)^^.
4 ( t -- 2 ) '

1

—-2t + 8lnt-2

Bai 8: Tinh H =

f

,

tdt

8
(t-2)^j

=f '

M-y/l+t'

,
=ln

. c , voi t =

a)f(x) =

l-t^

d e d u a ve

X

,

I

b)f(x) =

cosx

1+sinx
sin x(l+cosx)

Giai

+ Vx' -4x + 8

. , f dx
rcosxdx
fcosxdx
I= I
= I
5— =
r-r
""cosx "'cos^x
•'l-sinD$t u = sinx ==> du = cosxdx, ta co:
6 : I1= - J ^ .

2-'Vl-u

x+I

l +u
1 + sin X
du = —In
+C = -ln
+C
2 1-u
2 1 - sin x
1+u;

b) D $ t t = t a n - = * d t =
2
+C

dx
f
,=
^1 + Vx + 2X + 2

.

^

l + t'

2t

D a t t + x = V x ^ + 2x + 2 = > x = - — ^
2(1-t)
J r(l-l)

= - l n l - t — + C, v 6 i t = V x - + 2 x + 2 - x
t

^dt

cos

2t

1+

t'-2t +2
fl^-^-dt:=

—dx = ——dx=?dx=

•
2t
l - t '
l a co smx = — ~ , cosx = - — — , suy ra
l + t^
l +t

Giai

32

2t

i)

l + V x ' + 2x + 2

-t^+2t-2
dx = ^ " " ; / d t , d o d 6 H =
2(1-t)-

, tanx =

p ^ l ; Tim nguyen ham aia cac ham so sau:

dl

D a t x + 1= l = > d x = - ^ , t a c 6 :
t
tf
t=
[_X
1 / I

l + t^

cong thuc ha bac de dua ve nguyen ham cac ham lugng giac co ban.

Giai

r G-

- . cosx
l + t-

Itr^ng giac c6 lien quan dao ham roi diing doi bien so'de tinh hoac dung

dx
j
(x + l)Vx^ +2x + 2

Bai 7: Tinh G =

2t

.

b. D^ng bac cao theo sinx, cosx, tanx: Bien doi ve d^ng chiira hai nhom ham

8
16
dt
t -2^(t-27

4J

,

D|[t t = tan—, ap dung sinx =

2
tich phan hijru t i theo an so t.

( t - 2 ) - + 4 dt, do do:
(t-2)=

1

Vi^t

l^ideS:NGUYEN H A M M O T S6 H A M SO Ll/QfNG G I A C

+C

Bai 6: Tinh F = J V x " - 4 x + 8dx

Dat t - X - A = - 4 x + 8

DWHKhang

1+

ft-+2t + K

i+t'
^

1

2^

+ - d t
1-t
X')

X

» l l t + 2 + i d t = — + 2t + in|t| + C = - t a n ^ - + 2 t a n - + l n tan— + C
2
"
2
2
2
2
: Tim hp nguyen ham ciia cac ham so
1
1 + sin x + cos

X

b)f(x)=-rVsin X


Cdiii ncing luyen thi DH - Nguyen ham - Tich phuii - So phi'rc ~ Trdn Ba Ha

ny_ TNHH MTV D VVH Khang Vi^

Giai

Suy ra:

2

X

a) Dat t = tan — => dx =
-di
2
1 +
dx

Do do:

r dt

+t

1 + sin x + cosx

1=

1 + t2t
1-t1+ - y +
,
1+t1 + t-

M

-dt
2 + 2t

2

sin'

(

2t
7t ^
1+t

dt=4

4t'

1 rl +2t'+l

1
b)Tac6: sin"* X
Dodo: .

cosx
13-lOsinx - ( 1 - 2 s i n - x)

dx

sinx

cos X
2sin- x - l O s i n x + 12

Ta c6: cot*x =

Dat t = sinx => dt = cosxdx, do do:
dt
cosx
-dx=
2 t ' - i a + 12
13-10sinx-cos2x

1=

t-3

1

2

t-3

-1 f dt
i(t-l\(t-'
2-'t--5t+6 " 9 2J(t-2)(t-3)
1

dt

sin X - 3

+ C=-ln
dt = - l n

sinx - 2
2 t-2
t-2;

(1 - s i i r x)cosxdx
;x.cosx
:
dxsin X
sinx
•'
sinx
Dat t = sinx => dt = cosxdx

b)I =

34

cos'

X .

dx =

fCOS^
fcos

I

cot"\

- + •

sin^x

sin'x

. ,

^m" x
sin'x

b) f ( x ) = -

cos2x

sin"x + cos-x
sin"* X

dx + fcot- X — ~ dx = -cotx
cot^x + C
J
sin"
V
sin" X
3
«5:Tim nguyen ham ciia f(x) = cof'x
Giai

COS" X

Giai
a) f(x) =

y-y + C

— cos''x - — cos''x + C
7
5

2t'

cosx

X

Dat t = cosx => dt =-sinxdx. Do do:

4J

L_2__L
t

— ^

sin

a) Jcos^ xsin' xdx = Jcos^ x(l - cos" x).sin xdx

I = - Jt'(l-t')dt = -J(t'-t')dt= -

2

+C

Giai

1
f dx
1
, =-tan
cor- +C
1 cot
sin'x 8
2 2
2 8
2
Bai 3: Tim nguyen ham cua cac ham so sau:
13-lOsinx-

b) f(x) =

:
l+ f

Do do

a) f(x) =

flzZL±l!\i

r
4

a) f(x) = cos''xsin^x

_L±tL^=f(i±i;i
X

f(l-t')'dt

Bai 4: Tim ho nguyen ham cua cac ham so:
2t

b)Datt= tan —=>dx =
^dt ; sinx=
2
1 + tdt
dx

sin X

-dx =

I = In I sinx I - sin^x + — sin^x + C

=:|nl + t + C = lnl + t a n - + C
X

cos'x^

sin X
Vi

+c

cos'' X _ (1-s'in" x ) ' _ l - 3 s i n ' x + 3sin''x + sin^x
sin*" X
sin'' X
sin^ X
1 3
3 ,
sin X

sin"x

sin x + cos'x
sin'^x

sin'' x
sm

Do do: cof^x =

- + C0t" X
X

sin
(

X

1
sin-

- + cot' \

sin'

X

cot^x^
X

sin" X

2
1
1 3
— + cot^ X . , + cot' x . — + — — +1
sin X
sin" X
sin x sin x


Cam nang lny^n thi DH - Nguyen ham - Tich p/ic'in - So phiic — Tran Bd Ha
Vai:

sin'* X

sin^ X

cot*^ xdx =

cos* x d x . Dat

+ cot^ X . — \ — , s u y ra:
sin" X

I u = cos' xdx => du = - 5 cos'' x sin xdx
dv = cos xdx => V = sin X

i = cos'^xsinx + 5 cos' X sin^xdx = cos^xsinx + 5

-t-COl X+ 1 dx
- c o t " X.sin" X
sin^ X
sin X

cos' x (1 - cos'x)dx

= cos'xsinx + 5 Jcos' xdx - 5 Jcos^' xdx

= -cotx + — cot''x - — cof^x + X + C
3
5

61 = cos'^xsinx

^ t^

^'"2x + ^ sin4x]

Bai 6: Tim ho nguyen ham cua cac ham so
a)f{x) =

sin2x

V a v I = — cos'^xsinx + — f — x + sin2x + — sin4x| + C
6
4 2
8

b) f(x) = —
—
sin x + cos X

1 + sin" X

Cdch khdc

Giai
a) Dat u = 1 + sin^x

du = 2sinxcosxdx = sin2xdx

cos*x =

f sin2x .
fdu ,

, ,
• '
+C
" dx = — = l n u + c = l n l + s n r X
•'1 + s i n X
•' u
,
cos2x
cos2x
b) Ta co:
— =
sin x + cos X , _ l 3 i „ 2 2 ^
Dat u = sin2x
f

cos2x

du

_ r du

5
_

1

r

1

^

1
+ u,

2
2V2

>/2+ sin2x
72 + u
1
+C
In
+C=
v/2-sin2x
N/2-U
2>/2

Giai
a) Taco:
4

- [ - + 2cos2x + - cos4x]
4 2
2
Suy ra: Jcos' x dx = ^ [ ^ x + sin 2x + ^ sin4x] + C

sin" 2x)cos2x

16

Bid8;Tinh:

x + - sin 2x + — sill 4x + — sin'' 2x + C
4
64
48

Isinxsin—sin—dx
J
2
3

X . X
1
3x .
X
X
I
X
3x . X 1
sinx.sin—sin—= — c o s — c o s — sin— = —cos—sin
cos—.sin
2
2
,t: 2
3 2
3
2
2
3 2

2
1
r3x
x
i
f3x
x^
1
— sin f x x^ - s i n
sin — + — - s i n
—+
—
—
—
—
12 —3)
4
4
1/
I2
3)
I2 3 ;
I 2
3)

b) f(x) = cos*x

ri + cos2xV ^_i_r,1 + 2cos2x + COS" 2x
cos''x •

+ (1 -

Giai

Bai 7: Tim hp nguyen ham ciia ham so
a) f(x) = cos^x

1 + cos4x^

5
3
Suy ra: Jcos* xdx = | i
dx
+
3cos2x
+
-cos4x
+ (1 - sin^ 2x)cos2x
'8

du = 2cos2xdx, suy ra .
_ 1 f

= - [1 + 3cos2x + 3cos^2x + cos''2x]
8

l + 3cos2x + 3

' s i n ' x + c o s ' X ^^"2 J j _ 1 ^2 " • ' 2 - u ' ' l^fl \yl2-u^^/2

In

/'l+cos2x

l + 2cos2x +

l+cos4x

. 5x
sin
6

. X
. llx
. 7x
sin — sm
+ sin —
6
6
6
-6
5x ^
x 6
llx 6
7x
1" D o d o : fsinxsin—sin—dx = —
—cos — + 6 ct)s — + — cos
cos—
6 11
6

7
6 J
T
3
x 3
5x 3
l l5x 3 6 7\
1= —cos
c o s — + —cos
cos — + L
2
6 10
6 22
6
7
6


- i^yny>in nuiii -

)iung luyirii

i icn jnuni - M)pniK

-

JYdn lid tia

~

C. B A I T A P T O N G H Q P V E N G U Y E N H A M
1. B A I T A P TV" L U A N

I5L

Bai 1: Cho f(x) = x V 3 - x

. T i m a, b, c de ham so F(x) = (ax^ + bx + c ) V 3 - X

la

m o t n g u y e n ham cua f(x).

"3V3

Vay F(x) =

+ 1'

3N/3 +

+ C = Oc^C = --

:J^[(5X + 3 ) V 5 ^

+ (5x + l ) / 5 x T T ]

T i m h a m so y = f(x) neu biet:

Giai

T a c o : D y = (-co; 3]
ax^ + b x + c _ -5ax-

h(12a - 3 b ) x +

6b-c

; f H ) = 2 ; f(I) = 4vn r(l) = 0

f ( x ) = ax+

Giai:
F{x) la n g u y e n h a m cua f{x) <=> F '(x) = f(x), V \ D y
o

2

2

Bai 2: Cho f(x) = cos'^x - sin''x. T i m nguyen ham

ax'

b

2

\

f(x) = —

X"

12

D o n g nhat ta c6: a = — ; b = - — '

,

b

Ta c6: f '(x) = ax + —

-5ax2 + (12a - 3b)x + 6b - c = 2x(3 - x), V \ 3

- + b + c = 2
F(\ cua f(x) biet F( —) = 0

lis giai thiet ta c6:

6

—
2

b+ c = 4

a + b = 0

Giai

f(x) = cos^x - sin''x = (cos^x - sin-x)(cos2x + s i i i - \
<=> f(x) = cos2x => F(x) =

Ti

\

6

2

Giai h ^ ta c6: a = 1, b = - 1 , c =

sin2x + C

F(-) = 0 < » - s i n - + C = 0 « C

3

73
=

'

Bai 5: T i m h a m so y = f(x) biet rling:

Vay F(x) = — sin2x

•

^

2

Bai 3. Cho f(x)

5

4

V3

1

1

-

f ' ( x ) = 4 N / x - X va f(4) = 0

4
1
V5x + 3 - V 5 x + l

Giai

. T i m nguyon ham F(x) cua f(x) biet F(0) = 0
Giai

X-

f(x)= 4 x - ' - x ^ f ( x ) = 4 . ^ -

X-

—

Dy = [-l;co)
f(x) =

Dodo:

V5x + 3 + V 5 x + l

F(x)

15
F(0) = 0 <:>

( 5 x + 3 ) ^ + ( 5 x + l)3

(5x + 3)2 4{5x + l ) ^

f(4) = 0 « ^ - 8
3
8

1
15

38

1

3

3

+c = o

+C=0 « C =
/-

x-

40

, Vgy: f ( x ) = _ x V x - y - y

- ^

J

+ C

3V3+I


Bai 6; Tim ham so y = f(x) biet r3ng: f (x) = ifx +
Giaj

+ 1 va f(l) = 2

.,'(^..

4
f'(x)= x-* + x ' + l = > f ( x ) = — + ^

f '(x) = 1 - <^os2x => f(x) = X - - sin2x + C
n
n
71
f / _ ) = — <=>
*U
4
4

+x+C

1
2

„

71

+C=4

V|iy:f(x) = x - - s m 2 x - f(x)=-xV^ + - x ' + C
4

4

+—
2

g^_g: H m hg nguyen ham ciia cac ham so'
7t ,
sin X - c o s x
b) l(x) = cot2(2x+ - )
a)f{x)=4
s i n x + cosx
Gi.ii

1 1 1
1
f(l) = 2 o - + - + - + C = 2<:=>c= ^ '
4 4 2
2
Vay f(x)= — x ^ + —x^ + — x^ + —
4
4
2
2

i)

Bai 7: Chung minh F(x) = I x | - ln(l + | x | ) Li mot ngii ven ham aia: f(x) = —
1+

ff(x)dx =

J

-

+ ^"^^) ^
sinx+cosx

f^(^'"^

•'

b) f
Giai

1+x

1+x

Khix<0:F'(x) = -x +
1-x

1-x

= l(x)

„ .
Ay
A x - l n ( l + A.\
Tai x = 0: hm — = l i m

A.\->0'
A\-*0'
Ax
x

Ax

dx

BailO: Tim ho nguyen ham ciia cac ham so snu:

khix>()

a) i(x) = cof'x

1+x
Do d6:F'(0) = 0. VayFXx) =

-1

|f(x)dx = - - ( c o t ( 2 x + - ) ) - X + C
2
4

,
,.
ln(l + A x ) , ,
.
= 1 - l i m —^
L = \-] = Q

\v->o*

1 + sin^ X
Giai

hay: F X x ) = =f(x)
1+ X

=i\„ *5 J
fcos'x.
r(l-sin"\)' , .
a) j c o r x d x =
—dx=
\^—d(smx)
sin
x
•'sin'x
•'
cin^v
1
d ( s i n x ) - In sin X
Vsinx sm x sin x y

Vay F(x) la mot nguyen ham ciia f(x)

k\
'

X

cotx

b) f(x) =

khi x = ()

0

C

sin"(2x + - )
4

2x +
A)
Dodo: ff(x)dx = l f
sin'^ 2x +
4]

= f(x)

Khix>0:F"(x) = l -

_|„i^i,, ^ + ^.^^^X +

khi

X <

0

il-x

cotx

.
r
~ q-qX=

cosx
o

j;

,

CO.s

r

dx =

X

sill'"'X

sin' X
5

4sin'' x

+c

dX

<( 1 ^ sin"* x)
Bai 8: Tim ham so'y = f(x) biet rang f '(x) = taii\.sin2x va f( —) = —
4
4
Gi.ii

'1+sm^x dCsin**
-"sinxd
+ sin x)
x)
^sin x(l + sin x)

4QJ

f '(x) = tanx.sin2x •

smx
cosx

40

.2sinxcosx = 2sin x

C.:

sin' X

1 + sin X

-"sin x d + s i i V x )
d(sin'' x)
1 + sin x

= - lVsin
f —X-

+c
41


Ctim

naug

/iivi'ii

ihi DH - Nguyen

ficiiii - 'llcli

plum

- So phi'rc

-

Trdn

( 7v /NHH

Bd Ha

sin X - cosx
dx

£)at t = Inx => d t =
X +

v/2 1

4.

- COS!

d

dx
2

4

2

8

2

8

4;

fdt

-)

-=

^71

4

Dat t = 1 - x'' => d t = -Sx^dx <=> x^dx =

Sin

4

J=

sin

4

cos

1

-dx =

cosxcos(x + - )
- d cos

1

Bai 16: T i n h C =

(laii(\x

G iai

4

-

d(cosx)

r 2tdt

X +

X +
V

cosx

—

4y

71

cos

x+ -

4;

+c
54ilZ:

+t)"

+ C = ln

J J + t

i

^

1-tJ

dt

+ C

f x"dx

dx
V2x + 1 +V2X-1

k = i j(V2x + l - V 2 . \ - l ) d \

l_r

1-t

-Jo-txi

TinhD =

t

Gi.il

Dat t =

i

1+ t

r^ 1

dt

c =JO-t-)!

cosx

^

2

j=

•"(l-xVx

= V x => t2 = X = > 2tdt = dx
X +

In cosx - I n cos

Bai 13: T i n h k =

dt

B = - ^ j t ^ d t = -12|^t^+c=-^(i-x-^)yrv+c
33
dx
r ax

= In

72

+ C

Giai

. 71

J=

41n'x

71

Sin ~

4

+ C = -

8

Giai

. 71

dx

Jx'Vl -x''dx

Bai 15: T i n h B =

4

sin-

4t^

7t^

- — c o t — + — + C ^'fy)?
2 8 J

cosx.cos(- + x)

J=

i_

Jt'~

dx

Tinh J =

1

Vi^t

X I

V

Bai 12:

Khang

G i ai

Gi.ii

^/2-^y2cos

DVVH

dx

Bai 11: T i n h I = f ^ =

1=

MTV

J(2x + l)=d(2x + l ) - ^ J(2\ 1)

6 - •(2x + l)V2x

+ l - (2x - 1 ) 7 2 ^ "I : + C

# ) =

^i-18:

X -

1

X

= t + 1 =^ dx = d t

f i l l ] ) '-dt = ^ 1 2
2
2t-

3t-

4t'

1^

•+ C = -

dl

I

2(x-!)'

3(x-iy

4(x-l/

+ C

T i n h E = lsin^xcos''xdx
Giai

E --Jsin^x cos^xd(cosx) = -J(l - cos'\)i.os'Ad(cosx)
42

43


D a t t = cosx ^ E = -1(1 - t 2 ) f d t = 4 l ( - l ' + t'')til

E = - - t ^ + - P + C = - - cos'^x + - cus'x + C
5
7
5
7
,
Bai 19: Ti'nh I : Jsin x-Jlcos x - I d x

b) 0 9 * "

= X

=> d u = dx

d v = e^^dx =>

V

= -e"

Jxe-^dx = -xe-" + Je-'dx = -xe-" - e->^ + C = - ( x + I )e-'' + C
pai 22: D u n g p h u a n g phap lay nguyOn h a m t i i n g p h a n hay t i m h p nguyen

Gi.ii

ham ciia:

D a t t = 2cosx - 1 => d t = -2sinxdx

b) y = x2 ln3x

a)y = x s i n -

1
sinxdx = — d t
2

Giai
a) Dat u = X

d u = dx
X

X

d v = sin — d x => V = -2cos( —)
2
X

Hay

Bai 20:

I=-^(2cosx-l)V2cosx-l

Tinh

—
COS"

i C

dx

d v = x^dx =>

COS"

x

fe'dt = e ' + C = e"'^"^+C

ham ciia cac h a m so':
a) y = yjx Inx

b) y = x.e"
Giai

dv =

-N/X

d x =>

V

Bai 23: Dat L =/x"e«dx

Bai 2 1 : D u n g p h u o n g phap lay nguyCn h o n i l u n g phan hay t i m h p nguyen

a) D a t u = Inx => d u =

dx

a) C h u n g m i n h : In = x " ^ - n l i . - i , V n e N "

a) In-.= J x " V d x = > ( l „ . , ) ' = x"-'.e''
Ta c6: [x'^e" - nin-i]' = n . x " - ' . e" + e^x" - n.x" ' .
b) t = l x V d x

V =

f^/x I n x d x = — x V x i n x - — x V x + C '
J
-1

Q

-1

b) T i n h I2

Do do: x^e" - nln-i = Jcx'^dx = L (dpcni)

2
^
Vx hi xdx = — x\/x In X - - ^ \/xd.\

11 = xe" - lo = xe" - e" = (x - 1 ) 6 " + C
12 = x^e" - 2I1 = x^e" - 2(xe'' - e") + C
fc = (x2-2x + 2)e« + C

Bal24:

Tinh

xdx

P
sin X
Giai

44

X

fxMn(3x)dx = — I n 3 x - - f x ' d x + C = — I n 3 x — x ' + C
J
3
3J
3
9

,11 \

— d x =

X

dx

Dat t = tanx => d t =

COS" X

X

X

x
Gi.ii

f e

X

fxsin — d x = - 2 x c o s — + 2jcos — d x =-2\cos— +4sin— + C
2
2
2
2
2
dx
b) Dat u = ln3x => d u =

E>^t u = x => d u = d x

= e«. x"


( n TNHH MTVDVVH

Cdni >icin^ luyen ihi DH - Nyiiyeii ham - lichphim - Sophi'rc — Tran Bd Ha

dx

dv =

I = eMn(1

V = -cotx

sin"

fd(sin x )

—^— = - X
snr X

cot

X

+ col xdx

-x

Cut \

f
= eMn(l + e^) -

sm x

e'

dx = e>^ln(l + e^) -

+

ln(sin x ]
^ ,
dx

r
Bai29:

COS' X

sin^

Tinh A =

sin

X

X

-^dx

+ cos

X

Giai
Dat u = In(sinx)
dv =

cos"

Giai

= > d u = cotxdx

cos

XetB =

dx

V = tanx
X

sin"* X + cos' X

f CCOS
OS

-B=

- sin

X

J C O—
S x + sin

cosx
f—
-dx
cosx + s i n x

Bai 26: Tinh I =

Sin

X

+ sin

X

i:

r(sin X + c i ) s x )
•= hi I sinx + cosx
c o s x + SI n X

+ C

b) T i n h :

Giai

COS"
= COStt

rx'-2x-l

• + sina

COS" a

x(x-l-l)(x-l)

-b + c - - 2

a =1
o

b = l

-a = - l

cd(sina) _ cosa
cosx

1 .

+—sinaln
2

1 + sinx
1-sinx

+ C

b)

X--2X-1

dx =

dx

x(x--F)

c =- l
r dx

dx
X -I-1

•' X

Bai 28: Tinh I = jeMn(l + e^)dx

u = hi(l+e")
Dat
dv = e''dx
46

du=v=e^

-dx
1 + e^

x-1

(a-i-b-i-c)\"-(-(c-b)x-a

a-f b - f c = l

,
Dong nha't ta c6:

Gi.ii

c

-dx

c
x"-2x-l
a
b
^
= _ +
+
x(x"-l)
x
x-fl
x-1

dx

-"l-siirix

+ C

sin2x +

Giai

X

-d(cosa)

2N/2

x(x^-l)

(alah5ngs6)

f sin x. cos a + sin a cos x

^ T

, X--2X-1
a
b
Bai 30: a) T i m a, b, c de:
^
= - +
x(x--l)
X x-fl

,
a)

1=

dx

2 - s i n " 2x

sin2x-\/2

D o do: I = — (x + In i sinx + cosx I ) + C

Bai 27: Tinh I = f ^ ' " ( ^ + " )
•' cos" X

•

( s i n 2 x - V 2 ) ( s i n 2 x + V2)

T a c 6 : l + J = Jdx = x + C

cos

'

cos2x

---hi

X

-dx =

1

X

--dx = 2

d(sin2x)
•••X

cosx-sinx

cos2x

dx =

1

-dx
••cosx + s i n X

I-J =

X

1 - - s n r 2.\
Giai

Xet J =

X

Ta c6: A + B = jdx = X + C

In(sinx) ,
- — d x = tan X ln(sin x ) - x + C

cos" X

D o do:

+ ln(e>^ + 1) + C

:(e^+ l)ln(e^+ l ) - e ' ' + C

= -xcotx + In I sinx I ^ C
Bai 25: Tinh

J j ^ d x = eMn(l + c^) - p--<^l-±lL-^dx

+ C-) -

X

f xdx

Do do:

KhangViel

^ In 1 X

I + In I x + 1 1 - In I x - 1 1

-H

x(x-fl)

C = hi
x-1

+c

^ a i 3 l : C h u n g m i n h tren doan [-2; 2] li.nn so
F(x) = - | ^ ( 4 - x ^ ) '

la m o t nguvCn hhm ciia f(x) = 2x.

V4-x^


t

Cam nang luy^n thi DH - Nguyen ham - Tic/i pluiii - So phuc - Trdn Ba Ha

Vi^t

-a = -2

Ciai

Vx e (-2; 2) ta c6: F ' ( x ) =

TNHHMTVDVVHKhang

2 3
'.-(4-x") 3 2

F '(x) = ^('') Vx e R
2.\

2 a - b = 7 <=>

b =

- 3

b-c--4

c=

1

(4 COS" x - 3 ) s i n 2 x

F'(x) = 2x V 4 - X - =f(x)

g ^ :

C h o f(x) =
l-2sinx
Tim mpt nguyen ham F(x) ciia f(x) bict

F'(-2*)=

lim

= lini - ( x - 2 ) > y 4 - x -

x—J'

X+ 2

=0=f(-2)

Ciai

''-•-2' 3

f(x)=
F(2-)= l i m - ^
X-.2

X- 2

= l i m - ( x + 2)N/4-x- =0 =f(2)
3

=1

( l - 4 s i n - x)sin2x
„ ,, .
, . ^
; — — —
= (l + 2 s i n x ) s i n 2 \
l-2sin2x

<:> f(x) = sin2x + 2sin2xsinx = sin2x + C()sx - i.\is3x

Do do F(x) la mpt nguyen ham ciia f(x) treii [-2; 2]
Bai 32: Tinh

|f(x)dx = - — cos2x + sinx - — sin3x + C
2
3

f-^^ilil_dx
•"x" + 5 x + 6

F(0) = - - + C = 0<r>C= ^ 2
2

Ciai
4x + l l

.

Jx'+Sx + e

c2(2x + 5) + l ,
•'x-+5x + 6

2x+5
•'x-+5x + 6

^

r

dx

J x ' + 5x + 6

, r d ( x ^ + 5 x + 6) ^ / J
x" + 5x + 6

•'vx + 2

F(\

Vav F(x) = — cos2x + sinx - — sin3x + —
• ^ 2
2
2
Bai 36: T i m mot nguyen ham cua ham s.V f(\

x + 3,

sinx\/cosx

biet nguyen ham

tri^t tieu khi x = 7t.
Gi.ii

= 21n|x2 + 5x + 6 | + l n | x + 2 | + l n | x + 3 | + C

4

Bai 33: T i m mot nguyen ham F(x) cua lihm so f(x) = sin2x. e"*'''' biet r3ng:

|f(x)dx =J(cosx)^d(cosx) = - ^ - ^ — + C

F(f)-0

Jsin 2 x e « * ' Mx = - je""' "d(cos^ x) = -e'"^'" + C

F(x) = -e™'"'" + C = > F ( ^ ) = - l + C = O o C = l

F(x) = — cosx yjcosx
4

F ( ^ ) = - - H ) V H ) + c = - ^ 4 C
4
4

1

F(ir) = 0 < ^ C =

V a y F ( x ) = -e'="'''^ +1
Bai 34: T i m a, b, c de F(x) = (ax^ + bx + c)o^ la mot nguyen ham ciia ham so:
f(x) = (-2x2 + 7 x - 4 ) e - »
Ciai

+C

V | y F(x) =

4
cosx y/cosx

4

+—
4

5ai37: C h u n g minh r^ng F(x) = i ( x + Vl + x" t ln(x + Vl + x^)) la mot nguyen

F '(x) = (2ax + b)e-' - (ax^ + bx + c ) c - = (-ax^ + {2a - b)x - c)e-"
ham ciia f(x) = Vl + x"
48

49