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NGUYEN HAM HUU TY

BÀI TẬP NGUYÊN HÀM HỬU TỶ
(Dùng phương pháp hệ số bất định đưa về tổng )
x 2 − 5x + 9
1. ∫ 2
dx
x − 5x + 6
2.
2 x 2 + 41x − 9
∫ x 3 − 2 x 2 − 11x + 12 dx
3.

1
∫ x 3 − 2 x 2 + x dx

5x 2 + 6 x + 8

6.

∫ ( x − 3) ( x + 1)

7.


4 x + 11
∫ x 2 − 5x + 6dx

8.
9.

4.

2

∫ (x

1
2

+ 3x + 2

)

2

dx
2

5.

x4 +1
∫ x 6 − 1 dx

cot x

∫ 1 + sin

9

x

dx


dx
∫ x 4 − 4x 2 + 3

x2 −1
∫ x 4 + 2 x 3 − x 2 + 2 x + 1 dx
15.

dx

∫ x (1 + x )
4

2

x2

6x
∫ x + 1 dx

16.

∫ (1 − x )

x +1
dx
11. ∫
x(1 + xe x )

17.

∫ x( x

18.

x4 +1
∫ x 6 − 1 dx

10.

12.



dx

14.

3

x 4 − 6 x 3 + 12 x 2 + 6
∫ x 3 −6 x 2 + 12 x − 8 dx

13.

sin x + cos x
dx
3 + sin 2 x

2 2

dx

dx
10
+ 1)

HD:
1.

x4 +1
( x 4 + x 2 + 1) − x 2
1
x2
=
=

x 6 − 1 ( x 2 − 1)( x 4 + x 2 + 1) x 2 − 1 x 6 − 1



1
1 1
1 
= 


x −1 2  x −1 x + 1



x2
đặt t = x 3
x6 −1

2.
3.

2

x 2 − 5x + 9
1 
 1
= 1 + 3


2
x − 5x + 6
 x−3 x−2

2 x 2 + 41x − 9
A
B
C
=
+
+
3
2
x − 2 x − 11x + 12 x − 1 x + 3 x − 4


2 x 2 + 41x − 9 = A( x + 3)( x − 4) + B( x − 1)( x − 4 ) + C ( x − 1)( x + 3)

Gán x=1,x= -3 ,x=4 tìm A ,B ,C


1
1
A
B
C
=
= +
+
như bài 3 5.
2
2
x x + 1 ( x + 1) 2
x − 2 x + x x( x + 1)

4.

3

x 4 − 6 x 3 + 12 x 2 + 6
− 8x + 6
= x+
3
2
x − 6 x + 12 x − 8
( x − 2) 3
− 8x + 6
A
B
C
=
+
+
2
MS
x − 2 ( x − 2)
( x − 2) 3 tìm A,B,C


5x 2 + 6 x + 8

6.
7.
8.

9.

( x − 3) ( x + 1)
2

A
B
C
D
+
+
+
2
( x − 3) ( x − 3) x + 1 ( X + 1) 2

=

2

4 x + 11
A
B
=
+
x − 5x + 6 x − 3 x − 2
2

(x

1
+ 3x + 2

2

)

2

1 
 1
=

 x + 1 x + 2 

2

1
1
1 1
1 
= 2
=  2
− 2  10.
2
2
x − 4x + 3
x − 1 x − 3 2  x − 3 x − 1

(

4

6x 3

(

)

= 6 x 2 − x +1 −

x +1
11.

)(

)

6
x +1

x +1
e x ( x + 1)
=
x 1 + xe x
e x x 1 + xe x

(

)

(

đặt t = 1 + xe x

)



1
1

t −1 t

π
π


2 cos x − dx
2 cos x − dx
sin x + cos x
( sin x + cos x ) dx =
4
4


dx =
=
12..
đặt
2
3 + sin 2 x


π 
π 
2 + ( sin x + cos x )


21 + cos 2  x −   2 2 − sin 2  x − 
4 
4 




π

t = sin  x − 
4

13.

cot xdx
cos xdx
1
t8
=
=

với t=sĩn
1 + sin 9 x sin x 1 + sin 9 x t 1 + t 9

(

1 

1 − 2 
 x 

x −1
dx =
dx
2
x + 2x − x 2 + 2x + 1
1
1


 x +  + 2 x +  − 3
x
x


2

14.

)

4

3

t = x+

1
x


15.

16.

1
1
1
1
= 4 − 2 +
2
x 1+ x
x
x
1+ x2
4

(

)

x2

(1 − x )

2 2

dx =

1

(1 − x )

2 2

2

1



17.

18.

(

)

x x +1
10

=

x

1
1− x2

1
1 1
1 
= 

x − 1 2  x − 1 x + 1



1



(1 − x )

2 2

10

(

x9
x 10 + 1

)

=


1 1
1
2
+


2
2
4  ( x − 1)
( x + 1) ( x − 1)( x + 1) 

đặt t = 1 + x 10

x4 +1
1
x2
=

x6 −1 x2 −1 x6 −1
Nguyễn văn Phép
Gv. PTTH BÌNH MINH



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