9. Partial Fractions

a1. Introduction

A partial fraction decomposition is useful for computing integrals of rational functions, i.e ratios of polynomials. Let's start with an example:

Compute \(\displaystyle \int \dfrac{x^4+2x-16}{x^3+4x}\,dx\).

The Partial Fraction Theorem from algebra (which you will learn about in this section) guarantees that the rational function \(\dfrac{x^4+2x-16}{x^3+4x}\) can be written as \[ \dfrac{x^4+2x-16}{x^3+4x}=x-\dfrac{4}{x}+\dfrac{2}{x^2+4} \] Notice that the rational function has been written as the sum of a polynomial, \(x\), and two simpler rational functions, \(-\dfrac{4}{x}\) and \(\dfrac{2}{x^2+4}\) whose denominators \(x\) and \(x^2+4\), are the factors of the original denominator, \(x^3+4x\). Once you have this partial fraction decomposition you can easily do the integral: \[\begin{aligned} \int \dfrac{x^4+2x-16}{x^3+4x}\,dx &=\int \left(x-\dfrac{4}{x}+\dfrac{2}{x^2+4}\right)\,dx\\ &=\dfrac{x^2}{2}-4\ln|x|+\arctan\left(\dfrac{x}{2}\right)+C \end{aligned}\] Note: The last term was integrated using the trig substitution \(x=2\tan\theta\).

On the next page, we look at the form of the general partial fraction decomposition. That will allow us to compute the integral of an arbitrary rational function.