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LearnCambridge97092026–2027Paper 3 · Pure Mathematics 3Integration

Paper 3 · Module

Integration

5 subtopics≈ 225 min in total

Start the module: Integrating eˣ, 1/x and trigonometric functions Open graphing calculator

Integrating eˣ, 1/x and trigonometric functions
Reversing the new derivatives: $\int e^{ax+b}\,dx$ , $\int\dfrac{1}{ax+b}\,dx$ (the missing case $n=-1$ ), and $\int\sin$ , $\cos$ and $\sec^2$ , always remembering to divide by $a$ .
≈ 40 min
Integration using trigonometric identities
Some trigonometric expressions can't be integrated as they stand. Rewriting them with identities, such as $\cos^2x=\tfrac12(1+\cos2x)$ and $\tan^2x=\sec^2x-1$ , turns them into standard integrals.
≈ 40 min
Exact definite integrals and logarithms
Evaluating definite integrals exactly when the answer involves $e$ , $\ln$ or $\pi$ ; simplifying logarithmic answers; spotting $\dfrac{f'(x)}{f(x)}$ ; and integrating with partial fractions.
≈ 45 min
Integration by substitution
The reverse of the chain rule: replace part of the integrand by $u$ , change $dx$ into $du$ , change the limits, and turn a hard integral into an easy one.
≈ 50 min
Integration by parts
The reverse of the product rule: $\displaystyle\int u\frac{dv}{dx}\,dx=uv-\int v\frac{du}{dx}\,dx$ . How to choose $u$ , when to use it twice, and the trick for $\int\ln x\,dx$ .
≈ 50 min