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
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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.
To decompose the rational function
:
- Step 1 Use long division (if necessary) 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.
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.
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Evaluate the following by the method partial fraction decomposition.
Solutions
Q(x) is a product of linear factors some of which are repeated
[edit | edit source]
If
appears in the factorisation of
k-times 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.

We now simplify the fuction with the property of Logarithms.

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3. Evaluate

using the method of partial fractions.


Solution
Q(x) contains some quadratic pieces which are not repeated
[edit | edit source]
If
appears we use
.
Evaluate the following using the method of partial fractions.
Solutions
If
appears k-times then use

Evaluate the following using the method of partial fractions.
Solution