Integration by Partial Fractions

A rational function is a ratio of two polynomials, $\dfrac{P(x)}{Q(x)}$, with $Q(x) \neq 0$. It is called proper if the degree of $P(x)$ is less than the degree of $Q(x)$, and improper otherwise.

Every improper rational function can be reduced to a proper one by long division: dividing $P(x)$ by $Q(x)$ gives a quotient $T(x)$ and a remainder $R(x)$ with degree less than that of $Q(x)$, so that

with $\dfrac{R(x)}{Q(x)}$ now a proper rational function. This matters for integration because there is no single direct formula for a general rational integrand — but a proper rational function, whose denominator factors into linear and quadratic pieces, can be broken down into simpler pieces called partial fractions, each of which integrates using formulas we already know.

The decomposition pattern depends on how the denominator factors:

• Distinct linear factors $(x-a)(x-b)$: write as $\dfrac{A}{x-a} + \dfrac{B}{x-b}$.
• Three distinct linear factors: $\dfrac{A}{x-a} + \dfrac{B}{x-b} + \dfrac{C}{x-c}$.
• A repeated linear factor $(x-a)^2$: $\dfrac{A}{x-a} + \dfrac{B}{(x-a)^2}$.
• A linear factor together with a repeated linear factor: combine the single and repeated forms above.
• A linear factor together with an irreducible quadratic factor: a linear-over-linear term plus a linear-over-quadratic term. …