Techniques of Integration: A Comprehensive Guide with Examples, Slides of Differential and Integral Calculus

Download Techniques of Integration: A Comprehensive Guide with Examples and more Slides Differential and Integral Calculus in PDF only on Docsity!

TOPIC

TECHNIQUES OF INTEGRATION

TECHNIQUES OF INTEGRATION

1. Integration by parts

2. Integration by trigonometric substitution

3. Integration by miscellaneous substitution

4. Integration by partial fraction

g^ ^ x

f x

H x 

dx

g x

f x

A rational function is a function which can be expressed as

the quotient of two polynomial functions. That is, a function H

is a rational function if where both f(x) and g(x)

are polynomials. In general, we shall be concerned in

integrating expressions of the form:

DEFINITION

If the degree of f(x) is less than the degree of g(x), their

quotient is called proper fraction; otherwise, it is called

improper fraction. An improper rational function can be

expressed as the sum of a polynomial and a proper

rational function.

x 1

x

x

x 1

x

2 2

3



 



Thus, given a proper rational function:

Every proper rational function can be expressed

as the sum of simpler fractions (partial fractions)

which may have a denominator which is of linear

or quadratic form.

i i

a x b For each linear factor of the denominator, there

corresponds a partial fraction having that factor as the

denominator and a constant numerator.

Case 1. Distinct linear factor of the denominator

That is,

 

 

1 1 2 2 n n

a x b

N

a x b

B

a x b

A

g x

f x

where A, B, …..N are constants to be determined

dx

a x b

N

dx ...

a x b

B

dx

a x b

A

dx

g x

f x

1 1 2 2 n n

    

 







Thus, 

EXAMPLE: Evaluate each integral.



 

dx

x x x

x

4 5

3 2

3

   



  

 

2

0

2

2

1 2 3

4 1

2. dx

x x x

x x

EXAMPLE: Evaluate each integral.



 



dy

y y y

y

3 2

4 4

3



  



dx

x x x x

x

5 4 3 2

2

3 3

1

( )

2

ax  bx  c

For each non-repeated irreducible quadratic factor

of the denominator there

corresponds a partial fraction of the form.

Case 3. Non-repeated quadratic factor of the

denominator

n n n

n n

a x b x c

N a x b M

a x b x c

C a x b D

a x b x c

A a x b B

g x

f x

 

 

 

 

 



 

 



2

2 2

2

2

2 2

1 1

2

1

1 1 1

( 2 )

...

( 2 ) ( 2 )

( )

( )

where A, B, …..N are constants to be determined

Thus,

n n n

n n

a x b x c

N a x b M

a x b x c

C a x b D

a x bx c

A a x b B

dx

g x

f x

 

 

 

 

 



 

 



    2

2 2

2

2

2 2

1 1

2

1

1 1 1

( 2 )

...

( 2 ) ( 2 )

( )

( )

n

( ax bx c )

2

 

For each repeated irreducible quadratic factor

of the denominator there corresponds

a partial fraction of the form.

Case 4. Repeated quadratic factor of the denominator

n

ax bx c

N ax b M

ax bx c

C ax b D

ax bx c

A ax b B

g x

f x

2 2 2 2

where A, B, …..N are constants to be determined

Thus,

n

ax bx c

N ax b M

ax bx c

C ax b D

ax bx c

A ax b B

g x

f x

( )

( 2 )

...

( )

( 2 ) ( 2 )

( )

( )

2 2 2 2

 

 

 

 

 



 

 



   

EXAMPLE: Evaluate each integral.

dx

x x

x x x



 

 

2 3

5 3

( 1 )( 1 )

2 3





2 2 2

( 1 )

x x

dx