A. CÁC PH NG PHÁP L Y TÍCH PHÂNƯƠ Ấ
I. PH NG PHÁP PHÂN TÍCHƯƠ
Chú ý:
( )
b b b
a a a
f (x) g(x) dx f (x)dx g(x)dx
+ = +
� � �
Bài 1 Tính I=
22
2
1
xdx
x 7x 12− +
Gi i:ả
( ) ( )
2
2
x 7x 12 A B
f (x) 1 1
x 7x 12 x 3 x 4 x 3 x 4
−
= = + = + +
− + − − − −
Xét :
( ) ( ) ( ) ( )
( ) ( )
A x 4 B x 3
7x 12 A B
x 3 x 4 x 3 x 4 x 3 x 4
− + −
−= + =
− − − − − −
( ) ( )
( ) ( )
A 9
A B x 4A 3B
B 16
x 3 x 4
= −
+ + − −
= =
− −
V y:ậ
( )
2 2 2 2
1 1 1 1
2
1
9 16 1 1
I 1 dx dx 9 dx 16 dx
x 3 x 4 x 3 x 4
x 9ln x 3 16ln x 4 1 25ln 2 16ln 3
� �
= − + = − +
� �
− − − −
� �
= − − + − = + −
� � � �
Bài 2 Tính I=
1
2
0
4x 11
x 5x 6
+
+ +
Gi i:ả
( ) ( ) ( ) ( )
( ) ( )
( )
1 1 1 1
0
0 0 0
A x 2 B x 3
4x 11 A B
f (x) x 2 x 3 x 2 x 3 x 2 x 3
A 3 3 1 9
f (x)dx dx dx I 3ln x 2 ln x 3 ln
B 1 x 2 x 3 2
+ + +
+
= = + =
+ + + + + +
=
= + = + + + =� � �
=+ +
� � �
Bài 3
2
0
I cos x cos 2x cos3xdx
π
=
Gi i:ả
( )
( )
( )
2
1 1 1
f (x) cos2x cos4x cos2x cos4x cos 2x cos 2x cos6x cos2x 1 cos4x
2 2 4
= + = + = + + +
2 2 2 2 2
0 0 0 0 0
1 1 1 1 1
I cos6xdx cos4xdx cos2xdx dx sin 6x sin 4x sin 2x
4 4 6 4 2 2 8
π π π π π
� � π π
� �� �
= + + = + + + =
� �
� � � �
� �
� �
� � � �
Bài 4 tính
2
0
sinx
I dx
cos x sinx
π
=+
( ) ( )
( )
( )
22
0
0
A B cos x A B sinx
sinx sinx cos x
f (x) A B
cos x sinx cos x sinx cos x sinx
1
A1 sinx cos x 1
2I 1 dx x ln cos x sinx
12 cos x sinx 2 4
B2
ππ
− + +
−
� �
= = + =
� �
+ + +
� �
= −
− π
� �
= − + = − + + =� �
� �
+
� �
= −
II. PH NG PHÁP Đ I BI N SƯƠ Ổ Ế Ố
D ng 1ạ
+ Ch n ọ
1 2 1 2
a (t ) & b (t ) t ; t
x (t) dx '(t)dt
= ϕ = ϕ
= ϕ = ϕ
+
f (x)dx g(t)dt
=
+
2
1
t
b
a t
f (x)dx g(t)dt=
� �
+
2
1 x x sin t t (0; )
2
π
− =� �
+
2
1
x 1 x t (0; )
sin t 2
π
− =
Bài 5 Tính
2
2
2
2
0
x
I dx
1 x
=−
Gi i:ả
Đ t ặ
x sin t 0 t< 2
π
=
x 0 t 0
dx=cos tdt
2
x t
2 4
= =�
π
= =�
( )
2 2
2
x sin t cos tdt 1
dx 1 cos2t dt
cos t 2
1 x
= = −
−
( )
44
00
1 1 1 1
I 1 cos2t dt t sin 2t
2 2 2 8 4
ππ
π
� �
= − = − = −
� �
� �
Bài 6
2
2
2
3
dx
I
x x 1
=−
Đ t ặ
1
x 0<t<
sin t 2
π
=
2
1
x 2 sin t t
2 6 cos t
dx dt
sin t
2 3
x sin t t
2 3
3
π
= = =� �
= −
π
= = =� �
2
2
2
cos t
dx cos t
sin t dt dt dt
cos t
1 1
x x 1 1
sin t sin t
−
= = − = −
−−
6
3
6
3
I dt t 6
ππ
π
π
π
= − = =
D ng 2: ạ
D t ặ
t (x)
= ϕ
(a) dx '(t)dt
(b)
α = ϕ
= ϕ
β = ϕ
b
a
f (x)dx g(t)dt
β
α
=�
� �
Bài 7
6
2
0
cos x
I dx
sin x 5sin x 6
π
=+ +
Đ t ặ
sin x t
=
x 0 t 0
dt cos xdx
1
x t
6 2
= =�
=
π�= =�
( ) ( )
( )
1 1 1
2 2 2
2
0 0 0
1
12
2
0
0
dt dt 1 1
I dt
t 5t 6 t 2 t 3 t 3 t 2
t 3 10
I ln t 3 ln t 2 ln ln
t 2 9
� �
= = = − =
� �
− + − − − −
� �
−
= − − − = =
−
� � �
Bài 8
( )
e
2
1
dx
Ix 1 ln x
=+
Đ t ặ
ln x t
=
1
2
0
x 1 t 0 dx dt
dt I
x e t 1 x 1 t
= =�
= =�
= =�+
Đ t ặ
t tan u
=
2
t 0 u 0 1
dt du
cos u
t 1 u 4
= =�
=
π
= =�
4 4
2
4
20
0 0
1du
cos u
I du u
1 tan u 4
π π
π
π
= = = =
+
� �
.
III. PH NG PHÁP TÍCH PHÂN T NG PH NƯƠ Ừ Ầ
Chú ý:
T ừ
b b
b
a
a a
d(u.v) udv vdu udv u.v vdu
= + = −�
� �
b b
a a
b
b
a
a
I f (x)dx g(x)h(x)dx
u g(x) du g '(x)dx I= uv vdu
dv h(x)dx v
= =
= =�
−�
=
� �
Bài 9
2
0
x cos xdx
π
D ng ạ
b b
a a
P(x)sin xdx P(x)cosxdx
� �
� �
� �
� �
Đ t:ặ
u x du=dx
cos xdx v v=sinx
=
=
( )
2
2
2
00
0
I x sin x sin xdx x sin x cos x 1
2
π
π
π
π
= − = + = −
Bài 10
1
x
0
xe dx
D ng ạ
b
x
a
P(x)e dx
α
Đ t ặ
x x
u x du dx
e dx v v e
= =
= =
( )
11
1
x x x x
00
0
I xe e dx xe e 1
= − = − =
Bài 11
2
2x
0
I e sin 3xdx
π
=
D ng ạ
b b
x x
a a
e sin xdx e cos xdx
α α
� �
β β
� �
� �
� �
Đ t ặ
2x 2x
u sin 3x du 3cos 3xdx
1
dv e dx v e
2
= =
= =
V y ậ
2
2 2
2x 2 x 2 x
0 0
0
1 3 1 3
I sin 3xe e cos3xdx sin 3xe J(*)
2 2 2 2
π
π π
= − = −
Xét
2
2x
0
J e cos3xdx
π
=
Đ t ặ
2x 2 x
u cos3x du 3sin 3xdx
1
dv e dx v e
2
= = −
= =
V y ậ
2
2 2
2x 2x 2x
0
0 0
1 3 1 3
J cos3xe e sin3xdx cos3xe I(**)
2 2 2 2
π
π π
� � � �
= + = +
� � � �
� � � �
Thay (**) vào (*) Ta có
2 2
2x 2x
0 0
2
2x 2x
0
1 3 1 3
I sin 3xe sin 3xe I
2 2 2 2
13 1 3 3 2e
I sin 3xe sin 3xe I
4 2 4 13
π π
π
� �
� � � �
� �
= − +
� � � �
� �
� � � �
� �
� �
−
� �
= − =�
� �
� �
