Integration by Partial Fractions

Theorem
Let $\map R x = \dfrac {\map P x} {\map Q x}$ be a rational function over $\R$ such that the degree of the polynomial $P$ is strictly smaller than the degree of the polynomial $Q$.

Consider the primitive:
 * $\ds \int \map R x \rd x$

Let $\map R x$ be expressible by the partial fractions expansion:


 * $\map R x = \ds \sum_{k \mathop = 0}^n \dfrac {\map {p_k} x} {\map {q_k} x}$

where:
 * each of the $\map {p_k} x$ are polynomial functions
 * the degree of $\map {p_k} x$ is strictly less than the degree of $\map {q_k} x$ for all $k$.

Then:
 * $\ds \int \map R x \rd x = \sum_{k \mathop = 0}^n \int \dfrac {\map {p_k} x} {\map {q_k} x} \rd x$

This technique is known as integration by partial fractions.

Also see

 * Integration by Substitution
 * Integration by Parts