6.13 Integrating Using Linear Partial Fractions

This topic is BC only. Partial fractions is used when a proper rational function can be broken into simpler fractions whose denominators are linear factors. Instead of trying to integrate the original complicated fraction at once, you rewrite it as a sum of easier fractions, usually of the form ${\frac{A}{x-a} or \frac{B}{x-b}}$. After solving for the constants, each fraction can usually be integrated using logarithms or basic antiderivatives.

The main idea is decomposition: if the denominator factors into distinct linear terms, then the original rational expression can often be written as a sum of simpler rational pieces. After multiplying through by the common denominator, you solve for the unknown constants and then integrate term by term. This topic is more algebraic than conceptual, but it is very useful because it turns a difficult rational integral into a collection of standard ones.