Integration by partial fractions

Summary

Integration by partial fractions is a method to integrate rational functions. We write the rational function as a polynomial plus a sum of simple fractions. Each simple fraction has a known antiderivative.

Remarks

Over $\mathbb{R}$, the simple fractions have the forms $\frac{A}{(x - r)^k}$ and $\frac{B x + C}{(x^2 + p x + q)^k}$, with $p^2 < 4 q$. Their antiderivatives use powers, the logarithm, and the arctangent. For example, $\frac{1}{x^2 - 1} = \frac{1}{2} \Big( \frac{1}{x - 1} - \frac{1}{x + 1} \Big)$. So every rational function has an elementary antiderivative.

Explanations

L1 human

A complicated fraction of polynomials splits into a sum of simple fractions. We can integrate each simple fraction by a standard formula.