The document covers fundamental concepts of integral calculus, specifically focusing on anti-derivatives, integration techniques, and formulas. Examples illustrate the integration process and the use of integration by parts. It also includes a table of integration formulas and their applications.

1. INTEGRAL CALCULUS
I B.Sc Mathematics
18UMTC21
Mrs. P. Kalai Selvi ,M.Sc., M.Phil.,
Ms. M. Indira Devi, M.Sc., M.Phil.,

2. Anti-derivative
If f(x) is a continuous function and F(x) is the function
whose derivative is f(x), i.e.: 𝑭′
(x) = f(x) .
then:
𝒇 𝒙 𝒅𝒙 = F(x) + c;
where c is any arbitrary constant.

3. Notation:

4. Indefinite and Definite Integrals
Indefinite
Definite
( )f x dx
2
1
( )
x
x
f x dx

5. For example, to integrate 4x, we will write it as
follows:
4𝑥 𝒅𝒙 = 2x2
+ c , c ∈ ℝ.
Integral
sign
This term is called
the integrand
There must always be
a term of the form dx
Constant of
integration

6. 0
sinI xdx

 

 
00
sin cos
cos cos0
( 1) ( 1) 2
I xdx x
 

 
   
     

Another example with limit value:
( )f x( ) ( )F x f x dx ( )af x( )aF x( ) ( )u x v x( ) ( )u x dx v x dx 
aax  1n
x n  1
1
n
x
n


ax
eax
e
a
1
x
ln xsin ax1
cosax
a

cosax1
sin ax
a
2
sin ax1 1
sin 2
2 4
x ax
a


7. TABLE OF INTEGRATION FORMULAS
1
1
1. ( 1) 2. ln | |
1
3. 4.
ln

  

 
 
 
n
n
x
x x x
x
x dx n dx x
n x
a
e dx e a dx
a

8. 2 2
5. sin cos 6. cos sin
7. sec tan 8. csc cot
9. sec tan sec 10. csc cot csc
11. sec ln sec tan 12. csc ln csc cot
x dx x x dx x
x dx x x dx x
x x dx x x x dx x
x dx x x x dx x x
  
  
  
   
 
 
 
 

9. 1 1
2 2 2 2
13. tan ln sec 14. cot ln sin
15. sinh cosh 16. cosh sinh
1
17. tan 18. sin
x dx x x dx x
x dx x x dx x
dx x dx x
x a a a aa x
 
 
 
   
    
    
 
 
 

10. Integration by parts

11. Let dv be the most complicated part of the
original integrand that fits a basic integration
Rule (including dx). Then u will be the remaining
factors.
(OR)
Let u be a portion of the integrand whose
derivative is a function simpler than u. Then dv
will be the remaining factors (including dx).

12. x
xe dxFor example:
x x x
xe dx xe e dx  
x x x
xe dx xe e C  
u = x dv= exdx
du = dx v = ex

13. 2
sinx xdx
u = x2 dv = sin x dx
du = 2x dx v = -cos x
2 2
sin cos 2 cosx xdx x x xdx   
u = 2x dv = cos x dx
du = 2dx v = sin x
2 2
sin cos 2 sin 2sinx xdx x x x x xdx    
2 2
sin cos 2 sin 2cosx xdx x x x x x C    
For example :

14. Thank you…