Introduction

In this module we will discuss functions $\frac{P(x)}{Q(x)}$, where $P$ and $Q$ are polynomials with the degree of $P$ less than the degree of $Q$.  We call these proper (degree of top less than the degree of bottom) rational (poly over poly) functions. 

In order to find $\int\frac{P(x)}{Q(x)}\,dx$, we first see if we know the antiderivative, then we see if substitution will work, then we see if we may have an easy trig substitution.  If not, we will consider the integration technique of partial fraction decomposition, which is a technique for turning proper rational functions $\frac{P(x)}{Q(x)}$ into sums of simpler rational functions that can be more easily integrated.

We begin with an example of a proper rational function, on the left below.  Notice that by getting a common denominator on the right-hand side of the equation below, we get the left-hand side.  Using partial fraction decomposition, we can reverse this process, beginning with the LHS and determining the RHS. $$\displaystyle\frac{3x+4}{x^2+3x+2}=\displaystyle\frac{1}{x+1}+\displaystyle\frac{2}{x+2}$$ This allows us to integrate the LHS: $$\int \frac{3x+4}{x^2+3x+2}\,dx = \int \left( \frac{1}{x+1} + \frac{2}{x+2}\right )\,dx = \ln\lvert x+1\rvert + 2\ln\lvert x+2\rvert+C.$$

There are four general cases, depending on what happens when we factor the denominator $Q(x)$: