Integration by Substitution (Pure Mathematics 3 — 9709) — Mathematics

Introduction

Integration by substitution is the reverse-chain-rule technique that allows you to transform a complicated integral into a standard form you already know how to handle. In 9709 Paper 3, the examiner always supplies the substitution — your task is to carry it out correctly, change every part of the integral (including the variable of integration), and produce an exact answer. This technique appears regularly in Paper 3 and is often worth 6–8 marks, so fluency is essential.

Core Concept

The central idea is to replace the original variable (usually $x$ ) with a new variable (usually $u$ ) so that the integral collapses into a recognisable standard form.

The substitution process — four mandatory steps:

Write $u$ in terms of $x$ , as given in the question.
Differentiate to find $\dfrac{du}{dx}$ , then express $dx$ in terms of $du$ .
Replace every $x$ -expression and $dx$ in the integral with $u$ -expressions and $du$ .
Integrate with respect to $u$ , then (for indefinite integrals) substitute back in terms of $x$ .

For definite integrals, there is a crucial additional step: change the limits from $x$ -values to $u$ -values using the substitution formula, and do not convert back to $x$ at the end.

The formal justification uses the chain rule. If $u = g(x)$ , then:

\int f(g(x))\,g'(x)\,dx = \int f(u)\,du

This is valid because $\dfrac{du}{dx} = g'(x)$ , so $du = g'(x)\,dx$ .

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Introduction

Core Concept

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