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Bài tập nguyên hàm, tích phân

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I. Tìm nguyên hàm bằng định nghĩa và các tính chất
1/ Tìm nguyên hàm của các hàm số.



x


Cx
xx
++−










x
x
+

C
x
x
+−









x
x


x






x
x


C
x
x
x
++−








xxx
++

C
xxx
+++















xx


Cxx
+−
 




x
x



 
Cxxx
++−




x
x


Cxx
+−








x








Cxx
++








 

xx



 

xx
x




 

Cx
+−





Cxx
+−−



!

!


Cee
xx
+−




!






x
e
x

!







C
a
a
xx
++





!


Ce
x
+

+



2/ Tìm hàm số f(x) biết rằng
"#$


"

#$%




+−
x
x

"
xx

#$





−−
xxx

" 



+
x
#$






−++
x
x
x

"



#$ 




6"
&&'&


=−==
fff
x
b





++
x
x
II. MỘT SỐ PHƯƠNG PHÁP TÌM NGUYÊN HÀM
1.Phương pháp đổi biến số.
()*+

dxxuxuf ',-
./01*234
 34
dxxudt '
=⇒
 +
∫ ∫
=
dttfdxxuxuf ',-
BÀI TẬP
Tìm nguyên hàm của các hàm số sau:




dxx 




 x
dx

dxx






x
dx


+
xdxx




+
dxxx




xdxx 


+


+
dx
x
x




+
dx
x
x






+

 xx
dx

dx

x
x





+
dxex
x 




xdxx 



dx
x
x





gxdx


x

tgxdx




x
dx



x
dx



tgxdx


dx
x
e
x




x
x
e
dxe



dx
x
e
tgx





dxx 





 x
dx



dxxx 



+

 x
dx






 x
dxx


++


xx
dx


xdxx



dxxx 




+

x
e
dx


dxxx 


+
2. Phương pháp lấy nguyên hàm từng phần.
5644&#$**$7892:*$7;<=;+
∫ ∫
−=
dxxuxvxvxudxxvxu ''
>?
∫ ∫
−=
vduuvudv
#@A44"A&A##"A
Tìm nguyên hàm của các hàm số sau:


xdxx 


xdxx 


+
xdxx 



++

xdxxx 



xdxx 


xdxx 


dxex
x



xdx


xdxx 

dxx





x
xdx



dxe
x


dx
x
x




xdxxtg



dxx


+
dxx 



xdxe
x



dxex
x





+
dxxx 



xdx
x



xdxx 0


+
dxxx 


+
dx
x
x




xdxx 


TÍCH PHÂN
I. TÍNH TÍCH PHÂN BẰNG CÁCH SỬ DỤNG TÍNH CHẤT VÀ NGUYÊN HÀM CƠ BẢN:




 x x dx+ +

2.



 
 
e
x x dx
x x
+ + +


2.


x dx−

3.


x dx+



4.


  x cosx x dx
π
π
+ +

5.


 
x
e x dx+


6.



 x x x dx+

7.


  x x x dx+ − +



8.



  x cosx dx
x
π
π
+ +

9.



 
x
e x dx+ +


10.




 x x x x dx+ +

11.


  x x x dx− + +



12.



  A( ).

+

13.

2
2
-1
x.dx
x +

14.

!

   
A


− −

15.
 

5
2
dx
x 2+ + −

16.



  A
  
( ).
ln
+
+

17.




 A

cos .
sin
π
π

18.




0 A

.
cos
π

19.

 
 

! !
! !
dx



+

20.


 

! A
! !
.


+

21.



A
 +

22.

 

A
! !
ln
.

+

22.


A
 sin
π
+

24.



++



 dxxx
25.

−−






 dxxx

26.





 dxxx
27.







 dxx
28.
dx
xx







+




29.







dx
x
xx

30.


e
e
x
dx


31.



dxx
32.
dx
x
xx
e

−+



33.
dx
x
x













 



II. PHƯƠNG PHÁP ĐẶT ẨN PHỤ:
 1.

 

 xcos xdx
π
π

2.

 

 xcos xdx
π
π

3.




 
x
dx
cosx
π
+

3.


tgxdx
π


4.


 gxdx
π
π

5.


  xcosxdx
π
+


6.



x x dx+

7.



x x dx−


8.

 

x x dx+

9.





x
dx
x +



 

 

x x dx−

 





dx
x x +







dx
x+

 





 
dx
x x

+ +








dx
x +

 

 


  
dx
x+

 



x

e cosxdx
π
π

 



cosx
e xdx
π
π


18.




x
e xdx
+

19.

 

 xcos xdx
π
π



20.



x
e cosxdx
π
π

21.



cosx
e xdx
π
π


22.




x
e xdx
+


 

 

 xcos xdx
π
π




 

 xcos xdx
π
π

 



 
x
dx
cosx
π
+






tgxdx
π

 


 gxdx
π
π





  xcosxdx
π
+





x x dx+

30.




x x dx−

31.

 

x x dx+

32.





x
dx
x +

33.

 

x x dx−

34.






dx
x x +

35.

 
e
x
dx
x
+

36.

 
e
x
dx
x

37.

  
e
x x
dx
x
+

38.

 

e
x
e
dx
x
+

39.


 

e
e
x
dx
x x
+


40.



  
e
e
dx

cos x+

41.


 
x
dx
x+ −

42.


 
x
dx
x +

43.


x x dx+

44.




dx
x x+ +


45.




dx
x x+ −

46.


x
dx
x
+

 

 
e
x
dx
x
+


47.

 

e
x
dx
x

48.

  
e
x x
dx
x
+

49.
 

e
x
e
dx
x
+

50.


 

e

e
x
dx
x x
+


51.



  
e
e
dx
cos x+

52.

 

+

x x dx
53.
( )



  +


x xdx
π
54.



 x dx−

55.



 x dx−

56.




dx
x+


57.
dxe
x


+




58.




dxe
x

1
3
0
x
dx
(2x 1)+

 
1
0
x
dx
2x 1+



1
0
x 1 xdx−


 
1
2
0
4x 11
dx
x 5x 6
+
+ +



1
2
0
2x 5
dx
x 4x 4

− +

 
3
3
2
0
x
dx
x 2x 1+ +




6
6 6
0
(sin x cos x)dx
π
+

 
3
2
0
4sin x
dx
1 cos x
π
+



4
2
0
1 sin 2x
dx
cos x
π
+


 
2
4
0
cos 2xdx
π



2
6
1 sin 2x cos2x
dx
sin x cos x
π
π
+ +
+

 
1
x
0
1
dx
e 1+




dxxx 





π
 

+




π
dx
x
x



+




π
dx
x
x

 






π
dx
x
x




−+
+





 dx
xx
x
 

++





xx
dx


2
3 2
0
cos xsin xdx
π

 
2
5
0
cos xdx
π


4
2
0
sin 4x
dx
1 cos x
π
+



1
3 2
0
x 1 x dx−


2
2 3
0
sin 2x(1 sin x) dx
π
+

 
4
4
0
1
dx
cos x
π



e
1
1 ln x
dx
x
+


 
4
0
1
dx
cos x
π


e
2
1
1 ln x
dx
x
+

 
1
5 3 6
0
x (1 x ) dx−



6
2
0
cos x

dx
6 5sin x sin x
π
− +

 
3
4
0
tg x
dx
cos 2x


4
0
cos sin
3 sin 2
x x
dx
x
π
+
+

 

+






π
dx
xx
x



−+




xx
ee
dx
 

+





π
dx
x
x








π
π
dx
x
tgx
 






π
dxxtg



+






π
π
dx
x
xx
 

+
+




π
dx
x
xx



+




π
dx
x
xx
 


+




π
xdxxe
x



−+



dx
x
x
 

+
e
dx
x
xx






+






π
dx
x
x

1
2
0
1 x dx−



1
2
0
1
dx
1 x+

 
1
2

0
1
dx
4 x−



1
2
0
1
dx
x x 1− +


1
4 2
0
x
dx
x x 1+ +



2
0
1
1 cos sin
dx
x x

π
+ +

 
2
2
2
2
0
x
dx
1 x−


2
2 2
1
x 4 x dx−


2
3
2
2
1
dx
x x 1−




3
2
2
1
9 3x
dx
x
+

 
1
5
0
1
(1 )
x
dx
x

+



2
2
2
3
1
1
dx

x x −


2
0
cos
7 cos 2
x
dx
x
π
+



1
4
6
0
1
1
x
dx
x
+
+

 
2
0

cos
1 cos
x
dx
x
π
+




++




xx
dx


++


 x
dx










dx
x
xx
 
8
2
3
1
1
dx
x x +



7
3
3 2
0
1
x
dx
x+

 
3
5 2

0
1x x dx+



ln2
x
0
1
dx
e 2+

 
7
3
3
0
1
3 1
x
dx
x
+
+



2
2 3
0

1x x dx+

 

+



xx
dx

II. PHƯƠNG PHÁP TÍCH PHÂN TỪNG PHẦN:
B0*C)*D*EF0D*GH
4 #'        ' 
b b
b
a
a a
x d u x v x v x u x dx= −
∫ ∫
















IDa
̣
ng 1

 
ax
ax
f x cosax dx
e
β
α
 
 
 
 
 


  ' 
 

ax ax
u f x du f x dx
ax ax
dv ax dx v cosax dx

e e
= =
 
 
   
 

 
   
= =
 
   
 
   
   
 

IDa
̣
ng 2:
  f x ax dx
β
α

J
K

 
 
 

dx
du
u ax
x
dv f x dx
v f x dx

=
=



 
=


=


IDa
̣
ng 3:


 
 
 

ax
ax

e dx
cosax
β
α

LM
N
A4
K
HM
N
*
N
M
N
*D*E4
%




 
x
x e
dx
x +

2J
K




 
x
u x e
dx
dv
x

=


=

+

.%


 

 
x dx
x −

2J
K




 
 
u x
x dx
dv
x

=


=



%
   
  
 
      
   

      
dx x x dx x dx
dx I I
x x x x
+ −
= = − = −
+ + + +
∫ ∫ ∫ ∫
(M

N
*+





dx
x
=
+

.J
O
0D*PQ0D*
N
D2B
R
.;
N
B
N
(M
N
*+




 


 
x dx
x+

.J
O
0D*PQ0D*
N
DP
O
0D*E
O
H2J
K

 
 
u x
x
dv dx
x
=



=

+


Bài tập





e
x
dx
x

 


e
x xdx





 x x dx
+

 



e
x xdx


 




e
x
dx
x

 


e
x xdx

 



 x x dx
+

 



e
x xdx





 x c dx
π
+

 


 
e
x xdx
x
+

 



 x x dx
+

 



x xdx
π

π

13.



 x
dx
x

14.


x xdx
π


15.


x
xe dx

16.



x
e xdx
π


Tính các tích phân sau
1)




 dxex
x
2)





π
xdxx
3)





π
xdxx
4)





π
xdxx

5)

e
xdxx


6)


e
dxxx



7)



 dxxx
8)

+



 dxxx
9)


+



 dxex
x
10)

π

 dxxx
11)





π
dxxx
12)

+




π
dxxxx

×