partial fraction decomposition

A ratio of polynomials, like (3x + 5) / (x^2 - x - 2), looks hard to integrate as a single lump. Partial fraction decomposition is the trick of splitting it back into a sum of simpler pieces — fractions whose denominators are the basic factors of the original — each of which you already know how to integrate. It is the reverse of adding fractions over a common denominator, and it is the workhorse that makes every rational function integrable in elementary terms.

The recipe: first make sure the top has lower degree than the bottom (if not, do polynomial long division and set the polynomial part aside). Factor the denominator into linear and irreducible quadratic factors. Then write one term per factor: a linear factor (x - a) contributes A / (x - a); a repeated factor (x - a)^2 contributes A / (x - a) plus B / (x - a)^2; an irreducible quadratic (x^2 + bx + c) contributes (Bx + C) / (x^2 + bx + c). You solve for the unknown constants A, B, C by clearing denominators and matching coefficients, or by plugging in convenient values of x. Each resulting piece integrates to a logarithm, an arctangent, or a power.

This is the reason rational functions never need anything exotic — the result is always a combination of logs, arctangents, and rational terms, so a rational function always has an elementary antiderivative. It also underlies the inverse Laplace transform, where you break a transfer function into partial fractions and read off the time response term by term. The one place beginners stumble is forgetting that a repeated factor needs a term for every power up to its multiplicity, and that an irreducible quadratic gets a linear numerator, not just a constant.