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 $Q(x)$:

Partial fraction decomposition overview

  1. Distinct Linear Factors:
    The simplest case is where all the roots of $Q(x)$ are real and all are different, so that $Q(x)$ factors as a product of distinct linear factors. E.g., $Q(x)=x(x-1)(x+2)(x+7)$.

  2. Repeated Linear Factors:
    The next simplest case is where are the factors are linear, but some are repeated, such as in $Q(x)=x(x-1)^2(x+2)^3$.

  3. Non-repeated quadratic factors:
    When $Q(x)$ has complex roots, then we get irreducible quadratic factors. Case 3 is where these are all distinct such as in $Q(x)=x(x-1)(x^2+4)$.

  4. Repeated quadratic factors:
    The most complex case is where there are repeated quadratic factors, such as in $Q(x)=x(x-1)(x^2+4)^2$.