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:


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

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- + —


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
0

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;

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 =


.

-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


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


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



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


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

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) =


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
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 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 ) =


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 -


•'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

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:



-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



''>i-'• •

'•••tdu =

-I
du

J=

b) Dat X = asint-—
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" -

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:


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

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



(

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

du = 2(x + l)dx
•


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
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=

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
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

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

,
v = - ~ c o s S x
d u = — sin(ln x)dx
X

d v = dx
B = xcos(lnx)+

^

Vx-' + k

Dat u = X => d u = dx

+ x^

- je^dx = (x - 1

-

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 =

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


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

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

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

+ 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
• •

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 „

• +

•

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


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:

,

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


^ 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

•' 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

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:


^ 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
t2 + x 2 - 2 t x =

Dat X - a = t

•

•r.-n •••KV

+ 2x + 3

t=-3

^
1 t-+2t + 3
dt
=> dx = —
(t
+


=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)


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)-^



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 ^ .


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



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

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


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= -


r
4

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

_L±tL^=f(i±i;idx =
^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^

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


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 •


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

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


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

^

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-

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

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 =


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
d u = dx

d v = e^^dx =>
^ + 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



COS"

x

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

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



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

—^— = - 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:
d u = cotxdx

cos


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)

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


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 -


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