One of the basic concepts of calculus that are studied in the course “Numerical methods of economics.” More detailed information here https://ek.biem.sumdu.edu.ua/

1. Integral calculus. Indefinite
Integral

2. Plan
1. Indefinite Integral.
2. Basic Integration. Integration by substitution.
3. Integration by Parts.
4. Integration of Rational functions.
5. Trig Substitution.

3. 1. Antiderivative

4. Antiderivative

5. Indefinite Integral

6. Table of Integrals
1.   C
dx
0 2.  
 C
u
du
3.  
1
1
1





   


u
u du C 4.  
 C
u
u
du
ln
5. ( 0, 1)
ln
   

u
u a
a du C a a
a
6.  
 C
e
du
e u
u
7.  

 C
u
udu cos
sin 8.  
 C
u
udu sin
cos
9.  

 C
u
udu cos
ln
tg 10.  
 C
u
udu sin
ln
ctg
11.  

 C
u
du
u
du
ctg
sin2
12.  
 C
u
u
du
tg
cos 2
13. 









 C
a
u
C
a
u
u
a
du
arccos
arcsin
2
2
 
a
u
a 
 ,
0
14. 









 C
a
x
a
C
a
x
a
x
a
dx
arcctg
1
arctg
1
2
2
 
0

a
15. C
a
u
u
a
u
du






2
2
2
2
ln
 
0

a
16. C
u
a
u
a
a
u
a
du





 ln
2
1
2
2
 
0

a
1. Basic Integration

7. 2. Basic Integration

11. 2. Integration by substitution
Example 1

12. Integration by substitution
Example 2

13. 3. Integration by Parts
Suppose u(x) and v(x) are two continuously
differentiable functions. The product rule states
(in Leibniz's notation):
Integrating both sides with respect to x,
then applying the definition of indefinite integral,
gives the formula for integration by parts.

14. Since du and dv are differentials of a
function of one variable x,
Integration by Parts

15. We consider three groups of integrals, for which the
formula of integration by parts is used.
I group of integrals
Let Pn is polynomial function of n-order
Integration by Parts

16. Example
►  
 























 x
xdx
v
xdx
dv
dx
x
du
x
x
u
xdx
x
x
4
sin
4
1
4
cos
;
4
cos
3
4
;
1
3
2
4
cos
1
3
2
2
2
    





















  x
v
xdx
dv
dx
du
x
u
xdx
x
x
x
x
4
cos
4
1
;
4
sin
4
;
3
4
4
sin
3
4
4
1
4
sin
1
3
2
4
1 2
    













  xdx
x
x
x
x
x 4
cos
4
4
1
4
cos
3
4
4
1
4
1
4
sin
1
3
2
4
1 2
    






 C
x
x
x
x
x
x 4
sin
16
1
4
cos
3
4
16
1
4
sin
1
3
2
4
1 2
  .
4
cos
3
4
16
1
4
sin
4
3
3
2
4
1 2
C
x
x
x
x
x 










 

17. II Type of Integrals
  
 
dx
x
x
x
x
x
x
P k
a
n arcctg
;
arctg
;
arccos
;
arcsin
;
log
   
   
.
;
...
;
...
















 dx
x
P
v
dx
x
P
dv
dx
du
u
n
n
Example
►
 



























 x
x
x
x
x
v
dx
x
dv
x
dx
x
du
x
u
xdx
x 2
3
3
2
2
2
2
ln
3
3
;
1
ln
2
;
ln
ln
)
1
(








  dx
x
x
x
x ln
3
2
3



























x
x
v
dx
x
dv
x
dx
du
x
u
9
;
1
3
;
ln
3
2 








 x
x
x 2
3
ln
3
3 3 3 3 3
2
2 ln ln 2 ln 2 .
9 9 3 9 27
x x dx x x x
x x x x x x x x C
x
 
         
          
 
         
         
 
 

18. III Type of Integrals (cyclic integrals)
 
   
.
...
;
...
;
cos
,
sin













 dx
v
dx
dv
dx
ae
du
e
u
dx
bx
bx
e
I
ax
ax
ax
►  

















bx
b
v
bxdx
dv
dx
ae
du
e
u
bxdx
e
I
ax
ax
ax
cos
1
;
sin
;
sin



 bxdx
e
b
a
bx
b
e ax
ax
cos
cos 















bx
b
v
bxdx
dv
dx
ae
du
e
u ax
ax
sin
1
;
cos
;
.
sin
sin
cos 








  bxdx
e
b
a
bx
b
e
b
a
bx
b
e ax
ax
ax
.
sin
cos 2
2
2
I
b
a
bx
e
b
a
bx
b
e
I ax
ax




 
 



 .
cos
sin
sin 2
2
C
bx
b
bx
a
b
a
e
bxdx
e
I
ax
ax


19. 4. Integration of Rational functions

21. Substitution

23. Polynomial Division

24. Example

25. Partial Fraction Expansion

31. Completing the Square