Partial fraction decomposition

Splitting a rational function into a sum of simpler fractions, each easy to integrate on its own.

Partial fraction decomposition rewrites a rational function $\frac{P(x)}{Q(x)}$ (with $Q$ factored) as a SUM of simpler fractions — one term per distinct linear factor $\frac{A}{x-p}$ , one term per power of a repeated linear factor $\frac{A}{x-p}+\frac{B}{(x-p)^2}+\cdots$ , and one term per irreducible quadratic factor $\frac{Ax+B}{x^2+c}$ — each of which integrates directly to a log or an inverse-trig function.

Examples

$\dfrac{1}{(x-1)(x+2)}=\dfrac{A}{x-1}+\dfrac{B}{x+2}$ , solved for $A,B$ by clearing denominators and matching coefficients (or the cover-up method).