How Do You Use Partial Fractions to Integrate a Rational Function?

Use partial fractions to rewrite a rational function as a sum of simpler fractions, then integrate each term separately. Fractions with linear denominators usually produce logarithmic terms.

This is Mathematics: Analysis and Approaches HL content from AHL 5.15, building on decomposition from AHL 1.11. The example below uses an analytic method suitable for Paper 1.

The Method

1. Check that the integrand is a proper rational function: the numerator must have lower degree than the denominator. If it does not, perform polynomial division first.
2. Factor the denominator. For two distinct linear factors, write $$\frac{P(x)}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}.$$