Integration of Standard Forms (P3 §3.5) — Mathematics

Introduction

Integration by reverse differentiation (also called inspection) exploits the fact that integration is the inverse of differentiation. At AS level you met simple cases such as $\int x^n \, dx$ . In Pure Mathematics 3 the syllabus extends this idea to six specific standard forms involving the linear argument $ax + b$ . Mastering these forms is essential: they appear as stand-alone marks, as components of integration by parts or substitution, and inside differential equations. Every mark you lose on integration in P3 often traces back to mishandling the chain-rule factor $\frac{1}{a}$ .

Core Concept

When a function $f(x)$ is composed with a linear function $ax + b$ , the chain rule tells us that

\frac{d}{dx}\bigl[F(ax+b)\bigr] = a \cdot f(ax+b)

where $F' = f$ . Dividing both sides by $a$ and integrating gives the key principle:

\int f(ax+b)\,dx = \frac{1}{a}F(ax+b) + c

This single idea underpins all six standard forms below. Note that $a \neq 0$ throughout, and $c$ is an arbitrary constant of integration.

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Introduction

Core Concept

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