It highlights the methods to the problems of Integration

1. Integration methods
Dr. Pankaj Das
Mail.id: pankaj.iasri@gmail.com

2. Strategy of integration
• As we have seen, integration is more challenging than differentiation.
• In finding the derivative of a function, it is obvious which
differentiation formula we should apply.
• However, it may not be obvious which technique we should use to
integrate a given function.
• The main challenge is to recognize which technique or formula to use.
• A prerequisite for strategy selection is a knowledge of the basic
integration formulas.

3. Strategy of integration
• Once armed with these basic integration formulas, you
might try this strategy:
1. Simplify the integrand if possible.
2. Look for an obvious substitution.
3. Classify the integrand according to its form.
4. Try again.

4. Strategy of integration
• Sometimes, the use of algebraic manipulation or trigonometric
identities will simplify the integrand and make the method of
integration obvious.
Here are some example
   
1
x x dx x x dx
  
 
2 1
2
2
tan sin
cos sin cos sin2
sec cos
 
       
 
  
   
d d d d
2 2 2
(sin cos ) (sin 2sin cos cos )
(1 2sin cos )
x x dx x x x x dx
x x dx
   
 
 


5. Indefinite integral:
4
12 x
y e

4
4
4
12 12
4
3
x
x
x
e
z e dx C
e C
  
 

z ydx
 
Example 1.
Indefinite integral can be represent as

6. Example 2. 12 sin 2
y x x

2
12 sin 2
1
12 sin 2 cos2
(2) 2
3sin 2 6 cos2
z x xdx
x
x x C
x x x C

 
  
 
 
  


7. Example 3. 2 3
6
y x
x
 
2
2
3
3
3
6
3
6
6
3ln
3
2 3ln
z x dx
x
x dx dx
x
x
x C
x x C
 
 
 
 
 
  
  

 

8. Definite integral
Determine the definite integral in each case as defined below.
b
a
I ydx
 
Example 1: Let our problem
0
sin
I xdx

 

 
0
0
sin cos
cos cos0
( 1) ( 1) 2
I xdx x
 

 
   
     


9. Example 2.
Let our problem
1
2
0
8 x
I xe dx

 
 
   
1
2
0
2
1
2 0
2 0
2
8
8 2 1
(2)
2 2(1) 1 2 0 1
6 2 1.188
x
x
I xe dx
e
x
e e
e


 


  
    
   


10. Manipulate The Integrand
Algebraic manipulations (rationalizing the denominator, using trigonometric
identities) may be useful in transforming the integral
into an easier form.
Here are an example
2
2
2
2
1 1 cos
1 cos 1 cos 1 cos
1 cos
1 cos
1 cos
sin
cos
csc
sin

 
  





 
 
 
 
 



dx x
dx
x x x
x
dx
x
x
dx
x
x
x dx
x

11. Manipulate The Integrand
For instance, ∫ tan2x sec x dx is a challenging integral.
 If we make use of the identity tan2x = sec2x – 1,
we can write:
 Then, if ∫ sec3x dx has previously been evaluated,
that calculation can be used in the present problem.

12. Use Several Methods
Sometimes, two or three methods are required to evaluate an
integral.
 The evaluation could involve several successive
substitutions of different types.
 It might even combine integration by parts
with one or more substitutions.

13. 1. Linearity

14. 2. Function of linear function
This can be solved by two methods
• Inspection method
• Substitution method

15. Inspection method

16. Inspection method

17. Substitution method

18. Substitution method

19. Substitution method
Algebraic manipulations (rationalizing the
denominator, using trigonometric identities)
may be useful in transforming the integral
into an easier form.
 These manipulations may be more substantial
than in Step 1 and may involve some ingenuity.

20. Some more examples
• Let our problem,
• We can write as,
3
3
tan
cos
x
dx
x

3 3
3 3 3
3
6
tan sin 1
cos cos cos
sin
cos
x x
dx dx
x x x
x
dx
x


 

Example 1

21. Some more examples
• Substitute u = cos x:
• We can write as,
3 2
6 6
2
6
2
6
4 6
sin 1 cos
sin
cos cos
1
( )
1
( )
 



 


 
 



x x
dx x dx
x x
u
du
u
u
du
u
u u du

22. Integration by parts

23. Integration by parts

24. Integration by parts

25. Integration by parts

26. Integration by parts

29. Reduction formula
• Reduction formulae are the formulae that may be used repeatedly to express
the integral of a complicated function in terms of simpler one.
• The reduction formulae are generally obtained by the application of the rule
of integration by parts.
• Reduction formula is regarded as a method of integration. Integration by
reduction formula helps to solve the powers of elementary functions,
polynomials of arbitrary degree, products of transcendental functions and
the functions that cannot be integrated easily, thus, easing the process of
integration and its problems.

30. Formulas for Reduction in Integration
The reduction formula can be applied to different functions including
trigonometric functions like sin, cos, tan, etc., exponential functions,
logarithmic functions, etc. Here, the formula for reduction is divided into 4
types:
• For exponential functions
• For trigonometric functions
• For inverse trigonometric functions
• For hyperbolic trigonometric functions
• For algebraic functions

31. Generating a reduction formula: Definite
integral
• Integrands of the form and
sinn
x cosn
x
Using the integration by parts formula:
it is easily shown that:
udv uv vdu
 
 
1
n x n x n x
x e dx x e n x e dx

 
 

32. Generating a reduction formula: indefinite
integral
Writing:
then
can be written as:
This is an example of a reduction formula.
n x
n
I x e dx
 
1
n x n x n x
x e dx x e n x e dx

 
 
1
n x
n n
I x e nI 
 

33. Generating a reduction formula: indefinite
integral
Sometimes integration by parts has to be repeated to obtain the reduction formula.
For example:
1
1 2
1
2
cos
sin sin
sin cos ( 1) cos
sin cos ( 1)
n
n
n n
n n n
n n
n
I x xdx
x x n x xdx
x x nx x n n x xdx
x x nx x n n I

 



 
   
   




34. Generating a reduction formula: Definite
integral
When the integral has limits the reduction formula may be simpler. For example:
0
1
2
0
1
2
cos
sin cos ( 1)
( 1)
n
n
x
n n
n
x
n
n
I x xdx
x x nx x n n I
n n n I










 
   
 
   


35. Generating a reduction formula:
Similarly we can derive the following forms,
sinn
n
I xdx
 
1
2
1 1
sin .cos
n
n n
n
I x x I
n n



  
Similarly,
1
2
1 1
cos .sin
n
n n
n
I x x I
n n



 
cosn
n
I xdx
 

36. Generating a reduction formula:

37. Some important reduction formulas:

38. Example 1.