A Level Maths · Pure · Year 12 and 13
Integration starts as the reverse of differentiation and finishes with substitution, integration by parts and differential equations. Area questions are common on every board, and so are exact answers with ln or e.
AS content is usually taught in Year 12. The full A Level adds the Year 13 content.
The gradient function of a curve is given by $f'(x) = 1 - 6x$. Given that the curve passes through the point $(-1, 5)$, find the equation of the curve $y = f(x)$.
Then practise: skills to practise · open integration practice questions.
Every board examines the same A Level Maths content, set by the Department for Education. Only the paper it appears on and the section numbering differ.
| Board | Specification reference | Examined in |
|---|---|---|
| AQA 7357 | AQA section H | Paper 1, and Section A of Papers 2 and 3 |
| Edexcel 9MA0 | Edexcel topic Pure 8 | Paper 1 and Paper 2 |
| OCR A H240 | OCR reference 1.08 | Paper 1, and the pure sections of Papers 2 and 3 |
Three examples from the question bank. Every question comes with a step-by-step mark scheme (M, A and B marks) once you start practising.
Evaluate the definite integral $\int_{1}^{3} (3x^2 + 2x) \, dx$.
Mark scheme: in the 48-hour free trial.
Given that $\frac{dy}{dx} = 3x^2 + 4x$ and that $y = 10$ when $x = 2$, find an expression for $y$ in terms of $x$.
Mark scheme: in the 48-hour free trial.
Using the substitution $u^2 = 2x-1$, find $\int x\sqrt{2x-1} \, dx$.
Mark scheme: in the 48-hour free trial.
Questions are original and written in the style of Edexcel 9MA0, and cover content that AQA and OCR examine too. They are not past-paper questions.
These skills sit in the Pure III · Calculus unit of the UK A Level practice engine. Pick any mix of them, set the difficulty, and check each answer against its mark scheme.
Integration can come up on these papers (from each board's paper structure). We haven't mapped individual questions yet, so open a recent paper and look for it.
| Board | Recent papers that can test it |
|---|---|
| Edexcel 9MA0 | Paper 1: June 2024, June 2023, June 2022 Paper 2: June 2024, June 2023, June 2022 |
| AQA 7357 | Paper 1: June 2023, June 2022, November 2021 Paper 2: June 2023, June 2022, November 2021 Paper 3: June 2023, June 2022, November 2021 |
| OCR H240 | Paper 1: June 2024, June 2023, June 2022 Paper 2: June 2024, June 2023, June 2022 Paper 3: June 2024, June 2023, June 2022 |
Papers are © the exam boards; each page links to the official question paper and mark scheme. All A Level Maths past papers →
Yes. Integration is part of the A Level Maths content that every board examines: AQA section H, Edexcel Pure 8, OCR 1.08. It is assessed in AQA Paper 1, and Section A of Papers 2 and 3; Edexcel Paper 1 and Paper 2; OCR A Paper 1, and the pure sections of Papers 2 and 3.
Both. The AS content, usually taught in Year 12, is: Integrating xⁿ (n ≠ −1), including finding a curve from its gradient; Definite integrals and the area under a curve. The full A Level adds: Integrating eˣ, 1/x, sin x and cos x, and f(ax + b); Area between two curves or a curve and a line; Integration by substitution and by parts; Integration using partial fractions and trigonometric identities; Solving first-order separable differential equations and interpreting solutions in context.
Use it for a product of two different kinds of function, such as x eˣ or x sin x. Choose u to be the part that gets simpler when differentiated (usually the polynomial), except with ln x, where u = ln x.
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