A Level Maths: Integration By Substitution 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.
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 — foundational integration
- Differentiation — the chain rule is the differentiation analogue
- Further Integration — provides surrounding techniques
A Level Maths integration by substitution is the integral version of the chain rule. When you spot an integral that contains a function and (a constant multiple of) its derivative, substitution simplifies it dramatically. This is a powerful technique that handles many integrals that no other Year 1 or Year 2 method can.
You substitute $u = g(x)$, so that $du = g'(x) \, dx$, and rewrite the entire integral in terms of $u$ — including the limits if it is a definite integral. The standard pattern is $\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du$. You learn to spot suitable substitutions: for integrals containing $g(x)^n$ with $g'(x)$ elsewhere, or trig functions with their derivative as a factor, or expressions like $x\sqrt{x^2 + 1}$. For definite integrals, you change the limits using $u = g(x)$: if $x = a$ then $u = g(a)$, if $x = b$ then $u = g(b)$, and the answer comes out in $u$ without needing to back-substitute.
Integration by substitution 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: when $u = g(x)$, the differential is $du = g'(x) \, dx$ — you must replace $g'(x) \, dx$ with $du$ in the original integral; for definite integrals, EITHER change the limits to $u$-values OR back-substitute to $x$ and use the original limits — do not mix; if $g'(x)$ is not present (or only differs by a constant factor), substitution will not work cleanly; and after substitution, the original variable $x$ must NOT appear in the new integral — if it does, you have not made a complete substitution.