One may try to start by setting equal to either the numerator or denominator; in each instance, the result is not workable.
When dealing with rational functions (i.e., quotients made up of polynomial functions), it is an almost universal rule that everything works better when the degree of the numerator is less than the degree of the denominator. Hence we use polynomial division.
We skip the specifics of the steps, but note that when is divided into it goes in times with a remainder of Thus
Integrating is simple. The fraction can be integrated by setting giving This is very similar to the numerator. Note that and then consider the following:
In some ways, we “lucked out” in that after dividing, substitution was able to be done. In later sections we’ll develop techniques for handling rational functions where substitution is not directly feasible.