Partial Fractions

Proper rational functions

A proper rational function is one in which
the degree of the numerator is less than
that of the denominator.

 eg.   

All others are termed improper.
These can be simplified through long division.

Example

so

 

General Forms

 

 

Partial Fractions

Partial fractions are decomposed components of a fraction

since

 

The steps taken to find the partial fractions depend on the original function

 

Distinct linear factors

Example

   

First,write the denominator in factorised form:

Now set this equal to two partial fractions

Thus

This is an identity,

so A and B can be found by comparing co-efficients

Now pick values of x to find A and B

Substitute back into equation

Repeated linear factor

Example

so

comparing co-efficients

giving

 

Irreducible quadratic factor

This occurs if the discriminant of the denominator is less than zero.

Example

so

comparing co-efficients

giving

 

 

Rules

 

The degree of the numerator must be lower than than that of the denominator. If not, divide!

 

 

 

 

Example

 

 

Integrating rational functions

 

Example

Find the  integral

 

 

First, find the partial fractions

Now integrate

   

 

Example

Find the  integral

 

       

Now integrate

     

 

Example

Find the  integral

      

    

Now integrate

 

Example

Find the  integral

   

so

giving

Now integrate