Integration by Partial Fractions (Pure Mathematics 3 — 9709) — Mathematics

Introduction

Many rational functions — fractions of polynomials — cannot be integrated directly using standard forms. The technique of integration by partial fractions breaks a complicated rational function into a sum of simpler fractions that can each be integrated using known standard results. This is a core skill in Pure Mathematics 3 (9709) and appears regularly in both structured and unstructured exam questions, often combined with limits to produce exact values.

The partial fraction types you need are restricted to those established in topic 3.1, meaning the denominator factors are limited to:

distinct linear factors,
a repeated linear factor, and
a quadratic factor of the form $ax^2 + b$ (irreducible over the reals) paired with a linear factor.

Core Concept

Given a proper rational function $\dfrac{f(x)}{g(x)}$ , we write it as a sum of partial fractions and then integrate each term separately.

The three partial fraction structures (from topic 3.1):

Type 1 — Distinct linear factors in the denominator, e.g. $(ax+b)(cx+d)$ :

\frac{px+q}{(ax+b)(cx+d)} \equiv \frac{A}{ax+b} + \frac{B}{cx+d}

Each term integrates to a logarithm:

\int \frac{A}{ax+b}\,dx = \frac{A}{a}\ln|ax+b| + c

Type 2 — A repeated linear factor, e.g. $(ax+b)^2$ :

\frac{px+q}{(ax+b)^2} \equiv \frac{A}{ax+b} + \frac{B}{(ax+b)^2}

The second term integrates to a power:

\int \frac{B}{(ax+b)^2}\,dx = \frac{-B}{a(ax+b)} + c

Type 3 — A linear factor and an irreducible quadratic factor $ax^2+b$ :

\frac{f(x)}{(cx+d)(ax^2+b)} \equiv \frac{A}{cx+d} + \frac{Bx+C}{ax^2+b}

The quadratic term splits (after decomposition) into:

\int \frac{Bx}{ax^2+b}\,dx = \frac{B}{2a}\ln(ax^2+b) + c \qquad \int \frac{C}{ax^2+b}\,dx = \frac{C}{\sqrt{ab}}\arctan\!\left(x\sqrt{\tfrac{a}{b}}\right)+c

Key prerequisite: If the rational function is improper (degree of numerator $\geq$ degree of denominator), perform polynomial long division first to obtain a polynomial plus a proper fraction before decomposing.

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Introduction

Core Concept

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