For partial fraction expansion to be possible, the denominator must be of a higher degree than the numerator. If it is, it must be factored. For example:
Now we can make the numerators equal to and cross multiplying.
We can now multiply by the denominator.
can be solved for setting equal to a number that will cancel one term out.
We now know that
which can be proved by cross multiplication.
we can use partial fraction expansion to solve seemingly difficult integrals. For example, we just found that
This means that
This can easily be solved to find that