A Level Maths: Further Integration Topic Summary and Resources
Video Lessons
Watch alongside the worksheet for the full lesson experience, then test your understanding with the lesson questions.
Revision Notes
Handwritten notes summarising the key ideas for each lesson. Ideal for quick review before a test.
Exam Questions
Past-paper-style questions organised by topic, with full mark schemes.
- Standard Integrals of Form f(ax + b) (OCR)
- Integration Involving Trigonometric Functions (OCR)
- Integration of Rational Functions (OCR)
- Integration By Parts (OCR)
- Integration By Parts (Edexcel)
- Area Between a Curve and the y-Axis (OCR)
- Areas Involving Two Curves (OCR)
Drawn from OCR and Edexcel past papers but designed to be useful for students of all UK exam boards — including AQA and OCR MEI — unless a sheet is explicitly board-specific.
Before You Start This Topic
It will help if you are confident with the following:
- Integration — the standard reverse-power rule and indefinite/definite integrals
- Rational Expressions — needed for partial fractions integration
- Trigonometry — Compound and Double Angle Formulae}}: needed for trig integration
A Level Maths further integration extends Year 1 Integration to a much broader set of techniques: integrating standard functions of $ax + b$, integrating using trig identities, integrating rational expressions using Rational Expressions (partial fractions), and integration by parts. Mastery here unlocks Differential Equations and serious applied calculus.
You integrate standard functions of the form $f(ax + b)$ using a quick reverse-chain-rule reasoning — for example, $\int (2x+3)^4 \, dx = \frac{(2x+3)^5}{5 \cdot 2} + c$. You integrate trig expressions by recognising identities — $\sin^2(x)$ is rewritten using the double angle formula as $\frac{1 – \cos(2x)}{2}$ before integrating. You handle rational expressions by first decomposing into partial fractions, then integrating each piece (most reduce to $\ln$-type or power-type integrals). Integration by parts uses the formula $\int u \, dv = uv – \int v \, du$, with strategic choice of $u$ (often using LIATE: Logs, Inverse trig, Algebraic, Trig, Exponential) to make the second integral easier than the first.
Further integration is part of the Pure Maths strand of A Level Maths for AQA, Edexcel, OCR, and OCR MEI students.
Watch out for…
A few things to be careful with: the integral of $\frac{1}{ax + b}$ is $\frac{1}{a}\ln|ax + b| + c$ — do NOT forget the $\frac{1}{a}$ factor from the inner derivative; for trig integrations, the choice of identity matters — using $\cos(2x) = 1 – 2\sin^2(x)$ lets you integrate $\sin^2(x)$ easily, while $\cos(2x) = 2\cos^2(x) – 1$ helps with $\cos^2(x)$; integration by parts sometimes needs to be applied TWICE; and for $\frac{f(x)}{g(x)}$ with $\deg(f) \ge \deg(g)$, do polynomial division first before applying integration techniques.