Phương pháp tính tích phân và số phức phần 2

292

Tinh I =

•7

4
Jo

X

sin
x

X

+ (x + 1) C O S x ,
dx.

S i n X + COS X

DHkhoiA

-2011

Gidi
Ta

CO :

I =

(7

4
Jo

X

s i n x + (x + 1) cos x ,
n
dx =
X

s m X + cos
X

Jo
Ii=

pdx = x

Jo

X

X

cos

X

s i n X + cos

X

Jo

s i n X + cos
X

cos x

dx

X

~ 4

s m X + cos

- +1
4

du

-dx

Dat

u = xsinx -1- cosx

=> du = xcosxdx

X

= In u

2

= ln

V2

(7 X s i n X - 1 - ( x - f 1) cos X ,

1=

Tinh I =

X + X

s i n x + cos

-dx
X

_ 71

4

0

xcosx
X

u = l

x =0
D o i can

71
X = —

u =

4

V2

I2 =

Vay

93

•*
Jo
X sin X + cos x
rr 1 4- X s i n x ,
dx .

7t

V2 f 7:
2 v4

y

- In 1 = In
,

V2

^4-1

dx = — + I n —
4
4
2

cos^ x

Jo

BHkhoiB

-2011

Gidi
7C

Ta

I

CO :

Ii =

h =

\ X sin x

cos^ x
— = tan

^ cos

*

71

dx

dx =

* cos^

sin X dx
cos^ X

X

*

X

= V5

X

X

3 xsmx
cos X

u = X => du = dx
,
dv =

sinx ,
,
— dx chpn v =

cos

sin x
dx
=
cos^ X

cosx

X

134


- X Sin X,
Jo
3 C O S X dx =
3

294J

I =

Tinh I =

Jo cos X

X

2 Jo , s i n x - l

2K

Vay

COS

_ 271

dx

x

sinx + 1

n

d(sin x)

Jo s i r 2
sill X - 1

3

d(sin x) =

271

1

3

2

X- 1
sin X-1

sin

In

J

ln(2 - VS)
- 1

•0

+ X s i n Xdx
cos^ X

= V3+ — + ln(2-V3).

dx

- ^ l + x + x^ + Vx* + 3 x ^ + 1
Gidi

Dat

x = - t => dx = - d t

Vay

1 =

Suy r a : 21 =

Ta

x= - l

t = 1

X =1

t = -1

dx

-dt

^1

^

Doi can

1- t +
-1

+ Vt" + St^ + 1

•'-^ 1 + X + x^ + Vx^ + 3 x ^ + 1

[g(x) + g ( - x ) ] d x vdti g(x) =

1

1

+ X + X^

3x'

+1

CO :

g(x) + g ( - x ) =

1 + X + x^ + Vx"* + 3 x ^ + 1
1

- X + x^

(1

+ X + x^ + Vx'* + 3x2

+ X + x^ + Vx''

+ Vx" + 3x^ + 1 + 1
^

1 - X + x^ + A/X^+3X^+1
+ 3x^ + 1

_ X + x^ + Vx^ + Sx^

+ 1)

2(1 H-x^ + Vx^ +3x^+1)
(l + x 2 + V x ^ + 3 x 2 + l ) 2 - x 2
2(1 + x^ + V x ^ T s x ^ T l )
1 + x" + x" + 3x2 + 1 + 2x2 + 2Vx^ + 3 x 2 + 1 + 2x2 Vx^ + 3 x 2 + 1 - x2
2(1 + x2 + AM+3X2TI)
2 + 2x* + 4x2 ^ 2Vx*' +3X2+1 + 2x2 7 x ^ ^ 3 x 2 + 1

I + x2 + Vx^ + 3x2

^

1 + x^ + 2x2 + Vx^ + 3 x 2 + 1 + x2Vx* + 3 x 2 + 1
135


1+
+ Vx" + 3x2 ^ J
(i + x^+ V77377T) + (x2+ xnx2V77377T)
Do do :

Dat

21 =

J-i 1 +

•1

dx

•'0 1 +

^

X = t a n u => dx = (1 + tan^u)du

1 + tan^ u

I =

1

i + x'

dx

Jo l +

X^

fx = 0
D o i can
1
X =

n
4

Jo

295I T i n h I =

u =
U =

0
71
—

4

71

Jo

dx

^ x V x^ + 9
fiHAn

ninh

-1999

Gidi
Bat

t = Vx^ + 9 =>
X =

4

x = >/7
D o i can

= x^ + 9

1=

Vay

1=
t-3
= i l n
6
t + 3

Vay

Tinh I =

2tdt = 2xdx

t = 5
^

t = 4

dx

4

4

(t^ - 9)t

5

tdt

5

4

Esiel

4

8

" e l

dx

7J

dt

6

4

= - In - .

^ x y f ^ ^

6

4

dx

rl

1

+ X

+ V l + x^
HV Khoa hoc Qudn sU - 1998
Gidi

Ta

1 (1 + x) - V l + x^
1 (1 + xf --d + x

dx

CO :

• ' " ^ l + x + V l + x^

1 + x - V l + x^

fl

>

-1 u

^

(1

rl

2x

-1

dx

pi

- +1 dx-

-dx

-1

2x^

136


= -

- + 1 dx
dx = - ( l n | x | + x)
J

2J-1U

J-i

-1

X =

2 t d t = 2xdx

t=^

x =1

t

= .

Vay I , = r i i ^ „d—xax= =
J-i

97

1=

I

^

f f t-^4d t^ = 0

J
- ' ^^ 22 (( tt 22 - l )

2x2
2

r — ^ - 1 .

• ' - ^ l + x t V l + x^

-I

Tinh I

-1

2x2

e=

Nen

I

f i x V l + x^dx

Tinh l 2 =

Doi can

= 1

3

1

f

d x . B i e t fl(x) =

DHXdydung

- 1999

Gidi
ft
Ta

CO :

fllx)

A

=

V x ^

Vi

x e

V3

Vs

2

3

nen x > 0

T

xdx

2

Dat t = V l - x ^
X

=

X

=

Doi can

^

V8
3 ]

Si
-tdt

_

[-3

"

J l ( 1 - t )t

= 1 t =

2tdt = -2xdx

l

3

t = i
2
fg

dt

1,

t - 1
t +

^ = i l n ^
1 1
2
2

137


§9^

Vay

V8

_ J—

3 f
V3 ^

dx = - I n — .
2
2

T i n h 1 = j " ^e'^ - I d x .
Gidi
D a t t = V e " - 1 => t^ = e" - 1
V i t^ + 1 = e" => dx =

D o i can

Vay

X

2 t d t = eMx

2tdt
t^+l

= In 2

X =

t = Ve" - 1 = 1

0

t = Ve" - 1 = 0

ffin
in 2
2 I
f•1l tt^^d tt
Ve" - Idx = 2 f
=

1=1
Jo

Jot^+i

r l

2 | (d t - 2
Jo

1
0

dt

t^ + 1

D o i b i e n dat t = t a n u => d t = (tan^u + l ) d u
D o i can

fl

t = 1
t =0

n

u =—
4
u =0
• - ( t a n ^ u + 1)
du =

dt

ot^+l

0

'7 ,

tan^ u + 1

I=

fin 2 I
Ve" - Idx =

•4du
0

Tinh I = f

=4

0

Vay

[299I

= 2-2.^

4

=

2 - ^ .

2

dv

^ (2" - 9 ) V 3 - 2 ^ - "
Gidi

X

Ta

CO

:

22

I =

:dx =

* (2" - 9)73-2^""
Dat

t = V3.2" - 2

dx
* (2" - 9)73.2" - 2

=^ t ' = 3 . 2 " - 2

=> 2" - 9 =

2tdt
=> ^ ^ = 2"dx
31n2

t^ - 2 5

138


Doi can

Vay

1 =

X =

0

t = 1

X =

1

t = 2
2tdt

f2

.t
1
51n2
300

Tinh I =

•In

3 In 2

t-5
. — In
t + 5
I n 2 10

dt

In 2 J i t ^ - S ^

_9_
14

2" - 2"

r2

r2

4""

-dx.
-2
Giai

Dat

du
—
= (2''In 2

u = 2" + 2-"

D o i can

Vay

1=

X =

1

X =

2

2-'')dx va 4" + 4"" - 2 = (2" + 2"'f

5
u =—
2
17
u = —

2" - 2"

•2

- 4

1

•dx =
^
Ji 4" + 4"-" - 2

du _

1

- 4 ' In 2

2

du

In 2 Ji

- 4

2

17

1

, 81
•In—.
4 hi 2
25

u-2

•In
4 In 2
u + 2

5
2

Tinh I =

2e^''+e^''-l

.In 2

-dx.
+ e^" - e" + 1

JO

Giai
Ta

CO :

Vay

2e3- + e^^ - 1
+ e^" In 2

I = J-

26^"

(e^'^

+1 ~

- e ' +1)'

+6^''

+ 1

Jo

In 2
-

X

In 2

0

Tinh I

2x^ - 3x^ -

X

-1

•ln2d(e^=' + 6 2 " - e "

+ 6 ^ " - !

= ln(e="= + e^" - e'' + 1)

302

+1

e^" + e^'' -

„3x

e

, „2x

+e

„x

- e

+1)
, 1

1

rln2
JO

dx

, 1 1

=^"T-

+ Idx.

139


Dat
Ta

Gidi
t = 1 - X =>dt = -dx
CO :

2x^ - Sx^ - x + 1 = 2 ( 1 - tf - 3 ( 1 - t ) ^ - ( 1 - t ) + 1
= - 2 t ^ + at^ + t - 1

Doi can

Vay

1=

t = 0

1

t = 1

x = 0
X =

£ ^ 2 x ^ - Sx^ - x + I d x = ^ ^ - ( 2 t ^ - St^ - t + l ) ( - d t )

Jo

^2t^ - 3 t ^ - t + I d t = - I

S u y r a : 21 = 0 => I = 0 <=> I =

soil T i n h I =

Jo

dx

fO

^ 2 t ^ - St^ - t + I d t = 0.

"^^ 1 + V - x ( l + x )
Gidi
Ta

CO :

I =

0

dx

fO

1 + V - x ( l + x)

"'-^
1 +

1

dx

n

-

2

Dat

^
+ X

— + X = - s i n u => d x = - c o s u d u
2
2
2

Doi can

X =

71
U = —

2

u = 0

1
—

x = 0

Vay

1=

2

t
— cosuau
F
^
Jo
1 1
,
1 + J
sin'^ u
V4
4

1
p - ^ ^ d u =
Jo 2 + c o s u

J
0
Jo

1 -

2 + cos u

du

^-2p
^"
2
Jo 2 + cos u
Tinh J =
Jo

2 + cos u
u = 0

Dat

=> d u = '^^^
1+ e

t = tan—

Doi can

t = 0

71
U = —

t = 1

2

140


2dt

2 +
L a i dat

71

t = 1

a =—
6

[I f t a n ^

A/3

3 Jo

a +1

tan

a +1

da =

dx

I

2x + 3

•1

18

7i(9 - 2A/3)

18

+ x)

V-x(l
304| T i n h I =

a +l)da

"a = 0

rt = o

J =

D o do :

=> Sitan^

t = A/3 t a n a

Doi c a n

Vay

1 + t^

dx.

2 V2x - x^

Gidi
Ta

T

I =

CO

2x + 3
,
.
dx =
•'iV2x-x2

•1

2x - 2 + 2 + 3

2

V2x - x^

fi

rl

1

2x-2
>/2x - x^

2

* Tinh I i =

2x-2

•1

dx + 5

dx

dx
2x-x^

dx

2 V 2 x . - x^

Dat

u :

V2x - x^

=> u = 2x - x^

1
Doi c a n

2
X =

Vay

Tinhl2 =

^

1

•1 -2udu

Ii

2

*

u =

X = —

5

•1

u

=> 2udu = (2 - 2x)dx

A/3

u = 1
= - 2 73 du = - 2 u

dx

= -2 1-

dx

= 5

(x^ - 2x + 1)

Dat

x - 1 = sint

dx = costdt

Doi can

=

A/3-2

dx

^ Vl -

(X -

1
x =—
2
X = 1

1)^

"

6

t = 0

141


305|

Vay

5

«
"i

Do do :

I =

[-1

cos t d t

Vl -

= 5

0

- ^

dt = 5t

sin^ t

2x + 3

6

.dx = V 3 - 2 + ^ .

i >/2x Tinh I =

^ A/-3X^ + 6X + Idx .

Gidi
T a CO : f V-Sx^
Jo
Dat

+ 6x + I d x = ( ^j4-3(x-lfd\ 2 f , 1 - - (x - l)^dx
Jo
Jo V
4

Vs
2
— ( x - 1) = s i n t => dx = - = c o s t d t
2
V3

D o i can

Vay

1

X =

0

X =

3
t = 0

1=2

sm

2~
2
t - = cos t d t =

rO

V5

J

cos^ t d t

fO l + cos2t

. (1 + cos 2t)dt
tdt = • ^
V3 J

V3 J

2

(n

2 ( 1
t + - sin 2t
2

— +
3

V5
306| T i n h I =

''Vx^ - 2 x ^

S
—
4

3

+xdx.

0

DHThuy

M ~ A/2000

Gidi
V x ( x - D^dx = f ^ V x { l - x ) d x +

I =

X2dx-

1

2

=

i

2
x^dx +

rl

Jo
5

2

- - x 2

- x 2

1

o

+

5

0

0

2

4
. 1

fVxCx-Dd
i

4

x^dx -

C2dx

5

-2j

o

-X2

5

= 8.

142


Gidi
Dat

t = V x - 1
X = 2

t = 1

X = 1

t = 0

+1

•it^_+_t

Doi can

I =

308

fU^
0

1

= X -

2tdt = 2

t + 1

TInh I =

2 x - -

0 t + 1

2tdt = dx

• o

dt = 2

- t + 2 -

2

^
— dt = — - 4ln2 .
3
t + l j

Inxdx,

DH Khoi D

-2010

Gidi
•e (

T a CO : I =

1

h =

309

.2
«lnx
•Ji

X

I =

Tinh I =

Jo
f
•1

•e

h

Inx

xlnxdx

dx

dv = xdx

r

J

1

xdx =

2 Ji
2

2

+1
v 2 .

dx

D o i can

u = 0

X = e

u = 1

2
f

2

-A

3^
2 x - - I n x d x = 2 e"^ + 1
4

if

x = 1

1

udu =

(x +

V =

dx

•3 3 + I n x
1

dx

1

-Inx

u = I n x =:> d u =

h =
Vay

Inxdx = 2

u = I n x => d u =

Ii =

Dat

3^

xlnxdx

Ii =

Dat

2 x - -

1
-3.- =
2

-2

dx.

DH Khoi B - 2009
143


310

Ta

CO

: I =

f3

3 + lnx^
„
-dx = 3
(x +

Ii= 3

h =
Dat

f3

f3

ir

Gidi
dx
(x +

ir

+

•3

1

Inx
dx
(x + if

3

dx

•»i (x + if

(3 I n x
Ji (x + if

u = Inx

x + 1

-dx

du =

dx

dv

dx

=

(x + if

=>

f3
Inx
,
Inx
fS
dx
l2 =
r-dx = x + 1
Ji x(x + 1)
•"i (x + if
ln3
, 3
+ In4
2
r3 3 + l n x ,
3
ln3
, 3
1 =
-dx =
+In-.
J i l x + D^

4
4
2
Inx
-dx.

Vay

Tinh I =

V =

—

x + 1

ln3

fSdx
Jl

X

f3

dx

Jl X +

1

Jl x(2 + I n x f

DHKhd'i

B

-2010

Gidi
Dat

u = I n x => du =

—

D o i can

X

Vay

Tinh I =

Jo 1,2 + u

* (22 + uf

fe^

In^x

(2 + u)^

u = 1

= e

u = 0

=1

X

du = ln|2 + u

2 + u

0

3

dx.

xVlnx + 1

'I

X

ii

udu

•1

I =

1

l n 3 + - - ( I n 2 + 1) = I n
3
ill]

Bu hi - DI2005
Gidi
Dat

t^ = Inx + 1

t = Vlnx + 1

2tdt = —

va t^ - 1 = Inx

X

Doi c$n

t = 2

= e

t = 1

fx = 1
X

144


Vay

I =

In^x
'1 xVlnx + 1

= 2

3li

Tinh I =

,2t4

dx =

3

2tdt = 2

(t* - 2t^ + l)dt

Ji

76
15

+t
,5

- 2t^ +1

,

e Vl + S l n x . I n X

dx.
DH Khdi B - 2004
Gidi

Dat t = Vl + 31nx
Doi can

t =l
ft'-l

2

^ d t = 2 ^ ^ (t*-t2)dt = 9 5
3
9

116
3

135

e^Mx

Ins
ln2

= 1 + Slnx

t = 2

x = e
X = 1

I =

Tinh I =

=>

7771'
Dubil

-B/2003

Gidi
Dat

t = Ve" - 1

Doi can

I =

= e" - 1 ^ 2tdt = eMx,

+ 1 = e"

x = ln5
ft = 2
x =l n 2 ^ t = l
fin5
ln2

bl4| Tinh I =

=^

e^^'dx ^
,
dx =
^

X _ j

f2(t2

+l)2tdt
t

Jl

„
= 2 — +1
l3
J

2

20

e"

.ln3

•

=dx.
Du bi2

-AI2002

Gidi

Dat

t = e"

Doi can

+ 1 =>

dt = e''dx

' x = In3
_x = 0

rt = e^°^+l = 4
^

t = e°+l = 2
145


I =
°

dx =
VCe" + If

r4dt

.4

-1

t 2dt = - 2 t 2

t2

315| T i n h I = j j x^Va^ - x ^ d x

(a > 0).
£>// Su phqm

Ha Ngi + CD Hdi quan - 1999

Gidi
Dat

X = asint

Doi can

X

vdi t e

= a

x = 0

dx - acostdt

2
t = 0

f^l-cos4t,,
a^ r , s i n 4 t
dt = — t 8

I =

2 a^ sin^ t V a ^ d - s i n ^ t).a cos tdt

I =

316| T i n h I =

4 J0

2

2

a

71

16

dx

flnS

0

/ f y Q u d n 3/ - 7 9 9 7

Gidi
Dat

t = Ve" +1

Doi can

Vay 1 =
I =

I317I

= e" + 1

«

t = V2

0

X =

t = 2

ln3

X =

dx

•i"3

e

2dt

r2

^

„

2tdteMx

1,

=>

t-1

= ln

t + 1 V2

+

dx

In 3

dx = t^^'^^
^-l

(V2 + l)2

(V2 + 1)2
= ln-

0

T i n h va bi^n luan theo tham so difdng a, b : I =

a
dx
f* —
a2 + b 2 - 2 b x

DH Y Dicac TP.HCM

- 1980

146


Gidi
T a x e t f(x) =

a

MXD : D =

, CO

+ b
2b

Va2+b2-2bx

De h a m f(x) k h a t i c h t r e n [-a; a] t h i f(x) p h a i l i e n tuc t r e n [-a; a]
Suy r a
=>
Dat

[-a; a] c

a^+b^

-Qo;

a <

2b

(a - b f > 0

Va, b > 0

t = V a ^ + b^ - 2bx

=>

Do do :

ma a ;t b

•

'l«-b|-tdt _ - 1

dx

I =

a +b

V a ^ + b^ - 2bx
a+b

I = - t
b
Bien

1

b.t

|a-b|

b Ja+ b

dt

r

b^

|a-b|

2tdt = -2bdx

,.b|

t =

x = -a

2b

t^ = a^ + b^ - 2bx

't = |a-b

x = a

D o i can

a^ + b^

(a + b) - |a - b|

ludn : Neu a > b > 0 t h i I = 2
Neu b > a > 0 t h i I =

2a

dx

318| T i n h I =

' ( l + x " ) ! ^ l + x" '
DH Thai Nguyen

- 2000

Gidi
Ta

dx

f2
CO :
n+ l
X

Dat

t = nl— +1=>

+ 1

J

t " = — + 1

-n - 1

nt"-'dt = -nx-"-\dx

t" - Mt =

D o i can

X =

2

X =

1

dx
„n + l

t = - i ^ l + 2"
2
t = i2

147


0

Vay

I=

I = -

t°.t

2
•if2

t-^dt

2
'il2

Tinh I =

In

2

e'2x

!5/l + 2"

^

:dx.
Bdc/i khoa Ha. Noi - 2000
Gidi

Dat

u = Ve" + 1

Doi can

Vay

1=

=>

= e" + 1 =>

u =

x = 0

u = A/3

x = ln2

• In

2

p2x

2udu = e^dx

inZe^.e'^dx ,

- p = d x =

.V3

I = 2

- p = d x =

r ^3 (u^ - l)2udu

>/3

(u^ - l)du =

-2u
72

20

Tinh I =

^x^^Vl + Sx^dx.
CD Giao thong Van tdi -2000
Gidi

Ta

CO :

^x^Vl + 3x^ .x'^dx

Dat t = V l + 3x^ =>

21

t =2

1

-1 f 1
.t. — tdt

2t^

1 =

3

1

36

Tinh I =

= 1 + 3x* => 2tdt = 2 4 x M x
t =1

x = 0

X =

Doi can
Vay

I =

i

5

36

12

= ^
18

3

' Va^ - x ^ d x

'(t" - t 2 ) d t

Nen 1=

xi«ViT37dx = — ,

18

(a > 0).

148


Gidi

Dat

X = asint

Ta CO :

=> dx = acostdt

I=

x dx = a

1=

=a

x =0

X

2J
f ' V I ^ x dx =

2
t =0

a'^d-sin'' t)costdt = a''

1
2 ( l + cos2t)dt = - a ^ t + — sin 2t
2
0
2

I-la'
Vay

Doi can

2

2

cos^ tdt

na"

Tta

Jo

Gidi

Dat

x = asint

Doi can

a

Vay

2
0

a cos tdt

I=

7T

2' 2

dx = acostdt

t = ^
6
t = 0

X = —
X =

71

vdi t e

° V a ' d - s i n ^ t)

T cost ,
s
dt =
I'^ostl

6dt = ^

(vi 0 < t < 6
Vay

^^—^

dx

I=

Ji

=>

cost > 0, a > 0)

71

2x
DH Da Lat - 1999
Gidi

Dat

u = Inx => du =

dx

Doi can

x =e
X = 1

u =1
u =0
149


i

I

Vay

I=

fe V 2 + l n x ^
1 fi
V 2 + udu = —
dx = 2
2
1

(2 + u ) 2 d u

2x

1 2
= ±.f(2+
2 3

u)2

Cho so thuc b > ln2.

= -(3V3-2V2).
3

Tinh J =

/•In 10

Jb

p^dx

,

3/ V

va tim
„

lim J .
b->ln2

DH Quoc gia TP.HCM

- 1999

Gidi
Dat

u =: e" - 2

Doi can

X

du = eMx

= In 10

u = e^"i°-2 = 1 0 - 2 =

x=b
(•In 10

Vay

J=

u = e" - 2
e^dx

du

•8
K

—

3

-

4 - (e'' - 2)3

=

i

2
e''-2

Suy r a

lim J = lim — 4 - ( e ' ' - 2 ) 3
b^.ln2

25

= - . 4 = 6.
2

b->ln2 2

tan

Tinh I =

X

•dx.

Jo (4 cos X - s i n x) cos X

Gidi
Ta

CO

:

tan X
tan x
-dx =
dx
(4 cos X - sin x) cos x
Jo (4 - tan x) cos^ x

I

1 =

Vay

u = tanx => du =

Dat

X =

dx

Doi can

COS^ X

0

u =0

71
X = —

=>

u =1

4
tanx
-dx =
(4 cos X - sin x) cos x

•3

= -

du - 4
Jo

•1
•Q

-1

udu
4-u

3

f-1-

I

1

du
^
u-4J

= -1-4 In-.
4

-41n(u-4)

= -u
Jo u - 4

rr sin X + cos x ,
4
dx.
3 + sin 2x

|326|
26 T i n h I =

Jo

150

Gidi
Ta

CO :

Dat

7 s i n X + cos X ,
"
dx =
Jo
3 + sin 2x

I =

u : s i n x - cosx
"x = 0
X

Doi can

327]

X

4 - (sin

*

+ cos
X

X

2"
- cos x )

=> d u = ( s i n x + c o s x ) d x

u = 0

4
fO

Vay

sin

u = - 1

71

=

7

du

du

1 ,

-14-u'

I =

x + cos 5 x - cos 4 x

Tinh I =

-

u-2
u + 2

-1

= -ln3.
4

dx.

1 + 2 cos 3 x

8

Gidi
Ta

CO

:

x + cos 5 x - cos 4 x

I =

1 + 2 cos 3 x

- dx +
- - l + 2cos3x
8

*

Tinh I i =

dx

cos5x - cos4x
--

l + 2cos3x

dx

8

•dx

- - l + 2cos3x
8

7t

X =

Dat

t =

Doi can

=> d t = - d x

- X

71
X = —

8

Vay

Ii =

-s
-

tdt

-t(-dt)
l + 2cos3t

8

8

Suy r a : 2 I i =

n
- - 1
8

I, =

l + 2cos3t

xdx
+ 2C0S

8

•dx = 0
3x

- - 1
8

+ 2C0S

3x

•dx = 0
- - l + 2cos3x

151


* Tmhl2 =
Ta CO :

•J cos 5x - cos 4x

--

l + 2cos3x

dx

cos5x + cosx - 2cos3xcos2x va cos4x + cos2x = 2cos3xcosx

Suy ra : cos5x - cos4x = (cos2x - cosx) + 2cos3x(cos2x - cosx)
= (cos2x - cosx)(l + 2cos3x)
J,

cos5x-cos4x

1=

Vay

T

TT-

I2 = W

T

TTVay

:

Do

QO

1 + 2 cos 3x

=

(cos2x - cosx)(l + 2cos3x)
1 + 2 cos 3x

= cos 2x - cos x

f« cos5x-cos4x ,
(7
^ (cos 2x - cos x)dx

dx = »
8
Ll

i-^

l + 2cos3x

1
^
- sin 2x - sin x

-V2-V2

{2

17 x + cos5x - cos4x ,

-J2

r

]=

dx = — - V 2 - V 2 .

8

i-1

l + 2cos3x

2

328| Tinh I = rz xcos X + cos8x - cos7x dx.
1 + 2 cos 5x
Gidi

Ta

CO

:

I=

(4

X COS X + COS

8x - cos 7x

1 + 2 C O S 5x
X COS x
\x +
- - l + 2cos5x

dx

f7 cos 8x - cos 7x ,

^
dx
- i l l + 2cos5x

8

8

a

* Tinh I i =

X COS X

8

- i l l + 2cos5x

dx

•8

Ta

CO

f(x) = —
—
la ham so le do do I i =
1 + 2 cos 5x

xcosx

7
8

- - l + 2cos5x

dx = 0

8

cos 8x - cos 7x
dx
l + 2cos5x

* Tinhl2 =
8

Ta c6 :

cos8x + cos2x = 2cos5xcos3x va cos7x + cosSx = 2cos5xcos2x


Suy r a : cos8x - cos7x = (cos3x - cos2x) + 2cos5x(cos3x - cos2x)
= (cos3x - cos2x)(l + 2cos5x)
Vay

h =

17 (cos 3x - cos 2 x ) ( l + 2 cos 5x)
1 + 2 cos 5x

U

V2 +

V2

2

c o s 8 x-- ccosvx
o s 7 x ,,
f j x c o s x + cosox
J-il

329I T i n h I =

^ (cos3x - cos2x)dx
7t

I

71

fl- sin 3x —1 s i n 2x^ 8

dx =

V2 + A/2

V2

l + 2cos5x
tan

•

X

:dx.

cos x V l + cos^

X

Gidi
Ta

CO

:

tan

I=
Jo

Dat

tan

X

cos x V l + cos"

u = V2 + tan^ X

Jo

X

X

rdx
cos^ xV2 + tan^ x

= x + tan^x => 2udu = 2 i ^ ^ i L dx
COS^ X

x = 0

u = V2

n

D o i can

X = —

u = Vs

p/3 1

J3

4

Vay

1 =

^ — udu =

V3

du = u

42

U

30

Tinh I =

tan X
-dx.
Jo (4 cos X - sin x)cos X

Giai
71

Ta

CO

Dat

tan
(4 cos

X

n.

X

•dx
- sin x) cos x

u = t a n x => du =

dx
cos^ x

tan

X

-dx
( 4 - t a n x ) cos x
X =

D o i can

0

7:
X = —

u = 0
u = 1

4

153


331

Vay

1 =

'1
Jo

= -u

Tinh I =

1-3

0

4 -u

•1

udu

udu
u -4

+ 4 In u - 4

Vl - x^dx

(••1]

u- 4 + 4

.0

du = -

Jo

du + 4

—
Jo u -

= 41n--l.
4

bang each dat x = s i n t c6 ducrc hay k h o n g ?

Gidi
K h i d a t x = s i n t t h i - 1 < x < 1 n h i m g d day 0 < x < 3. V i k h o n g c6 gia
t r i t nao de 3 = s i n t
r3

Do do t i n h

x ^ V l - x^dx hkng each dat x = s i n t k h o n g ducfc.

cos X
,
^ ,
^dx.
V7 + cos 2x

332I T i n h I =

0

CD Hdi quan -

1999

Gidi
71

^
eosx
cos x
:dx = ^
Jo V7 + cos2x
-"o V 8 - 2 s i n 2 x

I =

=>

u = sinx

Dat

=> du = cosxdx

,
1
dx = — ^
V2 •'° ^4 - sin^

D o i can

n

X = —
X =

Dat

fi

I =

du = 2costdt

dx

u = 1
u = 0

4 - u^ = 4cos^t

t = ^
6
t = 0

u = 0

1

2
0

du

^J^TJ

V2 Jo

=>

u = 2sint
u = 1

D o i can

Vay

333I T i n h I =

r72eost ,
1
dt = ^ t
V2

71

6

V2 J 0 2 c o s t

6V2 '

sin2x

.dx.
1^
° Vcos^x + 4sin^x
DH Kiwi A - 2006

154


Gidi
Ta

CO

sin2x

:I=
cos

X +

4 sin

V2sin2x

I

2

Jo V 5 - 3cos2x
Dat

fx = 0

rdx

2(1 - cos2x)

,
dx

u =2

71
X = —

u =8

2

I=

«^J2_ _du_ _ V2
3 '2^

•2

334

7-

'1 + cos2x''|

u = 5 - 3cos2x => du = 6sin2xdx

Doi can

Vay

0

x

sin2x

Tinh I =

~ 3

- sin 2x + sin x ,
2
dx .
Jo Vl + 3 cos X
BH Khoi A - 2005

Gidi
Dat

t = Vl + 3 c o s x

=i>

7t
X = —

t =1

Doi can

X =

2
0

= 1 + 3cosx => 2 t d t = - 3 s i n x d x

t =2
t ^ - l

o sin 2x + sin x ,
2
dx =
0 V l + 3cosx

I =
O

-2

9

(2t^ + l ) d t = 9 Ji
9

3.

+ 1
f-—1 dt
.
3.

2

34

2t^

27

T I C K P H A N H A M C H L T A G I A T R IT U Y E T D O I

Phiicfng phdp

:

Muon t i n h

I =

|f (x)| dx

* X e t dau h a m f i x ) t r e n doan [a; b ] , de m d dau t r i tuyet doi.
* A p dung cong thufc :

rb
Ja

f ( x ) dx =

(.c
Ja

f ( x ) dx +

(.b
Jc

f(x) dx.

155


1 1
P35 1 Tinh I =
1

F3

-3

dx.

Gidi

1=

Vay

: I =

Ta

CO

f

X

3

, 3

x ^ - l dx =
A

-1

+

X
-3

V

-1
-3

X

3

(x^ - l)dx +
A

1

+

+ X

3

/ -1

-1

f a

(-x^ + l)dx +

f 3

(x^ - Ddx

>

X

X

3

J

1

1
1
1
1
44
- - +1+9 - 3 - - + 1 - - + 1 +9 - 3 - - +1=—
3
3
3
3
3

ISSGI
t

1

Tinh I =

x ^ - l dx = 44
|.2

x^ - 4x + 3 dx.

0

Gidi

Ta CO bang xet dau
x

0

-00

2

1

x^ - 4x + 3

f 1

x^ - 4x + 3 dx =

3

+00

-

0

+

Nen I =

(x^ - 4x + 3)dx +

(-x^ + 4x - 3)dx

1

^x^ - 2x2 ^ 3^

— + 2x2-3x

=2

0

Vay
337I

I =

Tinh

- 4x + 3 dx = 2.

=

Jo

X

-

2x + m dx.
Gidi

Dat

f(x) =

- 2x + m c6

A' = 1 - m

Khi m > 1 <=> A' = 1 - m < 0 => f(x) > 0, Vx e R
Do do

!(„,) =

(x^ - 2x + m)dx

x-^ - 2x + m dx =
1
X

I(m) =

2

3

= m- — .
3

+ mx

X

156


A' = l - m > 0
* Khi 0 < m < 1 t h i f (0) = m > 0
f (1) = m - 1 < 0
Phuong trinh fTx) = c6 hai nghiem X i < X2
Do do ta

0

CO

< Xi <

1<

vdi

X2

= 1 ± Vl -

x j , X2

ni

Hay ta c6 :
0

Xl

X2

0

f(x)

Nen :
f1
I(m) =

x^ - 2x + m dx =

X

2
X

=^1

''1

^

3

+ mx

f ^(-x'^
. „ +2 2x -

m)dx

2
+ X

3

- mx

2
• - x J + mxj + — m

3

= 2

xi =

•

X

0

The

(x''2 - o2x + m)dx +,

1 - Vl - m vac ta c6 :

U = - (1 - V l - m)[(1 - V l 3

m)2

- 3(1 - V l - m) + 3m + — m
J 3

i ( l - , ^ r ^ ) ( 2 m - l + Vr^) + - - m
Khi m < 0 t h i
Do do ta

CO

f (0) = m < 0
f (1) = m - 1 < 0

: Xl <

0< 1<

f(x) < 0 Vx e [0; 1]

X2

1

Nen I™ =

•1
(-X

+

2x - m)dx =

-X

2
+ X

- mx

2
=— m
3

T6m lai :
— m
3

neu m < 0

x^ - 2x + m dx = - (1 - V l - m)(2m - 1 + V l - m ) + - - m neu 0 < m < 1
3
3
2
m
neu m > 1
3

—

157


Isssj

T i n h t i c h p h a n I = £^

- X

dx .

DH Khoi

D - 2003

Gidi

|.2
. 0
f

x2-X

X

I =

Z

X

dx =
2>

rl

x)dx + J" (x^ - x)dx

'x^

x^

= 1.

I339I T i n h 1(a) = j ^ ^ x | x - a | d x .
DH Quo'c gia TP.HCM

- 1991 + DH Y duac TP.HCM

- 1996

Gidi
*

Khi a < 0 thi
Vay

1(a) =

x- a >0

Vx e [0; 1]

x|x-a|dx =

2 ^

fx'

ax
2

3

V

K h i 0 < a < 1, t a
X
X

1

J 0

_1

1

+

0
(•a

1(a) =

r\

xlx-a|dx+

Jo

a

" 3~ 2

CO :

a

0

- a
Vay

xlx-aldx

Ja

1(a) =

(-x^ + ax)dx +

(x^ - ax)dx

x 3" A*

ax''2

I(a) =

*

2

2 ^

ax

a^_ a

3

3

1

2^3

K h i a > 1 t h i x - a < 0 V x e [0; 1]
Vay

x^

1(a) = f \ - x 2 + a x ) d x =
Jo

ax-

a _1
2

3

158