Calculus/Integration techniques/Partial Fraction Decomposition
Suppose we want to find . One way to do this is to simplify the integrand by finding constants and so that
This can be done by cross multiplying the fraction which gives
As both sides have the same denominator we must have
This is an equation for so it must hold whatever value is. If we put in we get and putting gives so . So we see that
Returning to the original integral

= =
Rewriting the integrand as a sum of simpler fractions has allowed us to reduce the initial integral to a sum of simpler integrals. In fact this method works to integrate any rational function.
Contents
Method of Partial Fractions[edit]
To decompose the rational function :
 Step 1 Use long division to ensure that the degree of is less than the degree of (see Breaking up a rational function in section
1.1).
 Step 2 Factor Q(x) as far as possible.
 Step 3 Write down the correct form for the partial fraction decomposition (see below) and solve for the constants.
To factor Q(x) we have to write it as a product of linear factors (of the form ) and irreducible quadratic factors (of the form with ).
Some of the factors could be repeated. For instance if we factor as
It is important that in each quadratic factor we have , otherwise it is possible to factor that quadratic piece further. For example if then we can write
We will now show how to write as a sum of terms of the form
 and
Exactly how to do this depends on the factorization of and we now give four cases that can occur.
Q(x) is a product of linear factors with no repeats[edit]
This means that where no factor is repeated and no factor is a multiple of another.
For each linear term we write down something of the form , so in total we write
Example 1
Find Here we have and Q(x) is a product of linear factors. So we write Multiply both sides by the denominator Substitute in three values of x to get three equations for the unknown constants, so , and We can now integrate the left hand side. 
Exercises[edit]
Evaluate the following by the method partial fraction decomposition.
Q(x) is a product of linear factors some of which are repeated[edit]
If appears in the factorisation of ktimes then instead of writing the piece we use the more complicated expression
Example 2
Find Here and We write Multiply both sides by the denominator Substitute in three values of to get 3 equations for the unknown constants, so , , , and We can now integrate the left hand side. 
Exercise[edit]
Q(x) contains some quadratic pieces which are not repeated[edit]
If appears we use
Exercises[edit]
Evaluate the following using the method of partial fractions.
Q(x) contains some repeated quadratic factors[edit]
If appears ktimes then use
Exercise[edit]
Evaluate the following using the method of partial fractions.