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A Level Maths · Pure · Year 12 and 13

A Level Maths Integration revision

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.

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Same content for AQA, Edexcel and OCR
Step-by-step mark schemes
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What you need to know

Integration in Year 12 and Year 13

AS content is usually taught in Year 12. The full A Level adds the Year 13 content. Sitting AS exams or mocks? See the AS Level Maths checklist.

Year 12 (AS content)

Integrating xⁿ (n ≠ −1), including finding a curve from its gradient
Definite integrals and the area under a curve

Year 13 (full A Level only)

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

Worked examplemedium4 marks

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 $y = f(x)$.

Mark scheme and worked answer

M1 for integrating to find $f(x) = x - 3x^2 + c$.
M1 for substituting $x = -1$ and $f(x) = 5$: $5 = (-1) - 3(-1)^2 + c$.
M1 for solving to find $c$: $5 = -1 - 3 + c \implies c = 9$.
A1 for the final equation $f(x) = -3x^2 + x + 9$.

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Then practise: skills to practise · open integration practice questions.

Where it's examined

Integration on the AQA, Edexcel and OCR papers

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	Papers 1, 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

Practice questions

A Level Maths integration exam-style questions

Three examples from the question bank, one with its full mark scheme open. Every question comes with a step-by-step mark scheme (M, A and B marks) once you start practising.

Question 1easy3 marks

Evaluate the definite integral $\int_{1}^{3} (3x^2 + 2x) \, dx$.

Mark scheme: in the 7-day free trial.

Question 2medium4 marks

Given that $\frac{dy}{dx} = 3x^2 + 4x$ and that $y = 10$ when $x = 2$, find an expression for $y$ in terms of $x$.

Full worked solution Free, no sign-in

M1 for integrating $\frac{dy}{dx}$ to find $y = x^3 + 2x^2 + c$.
M1 for substituting $x = 2$ and $y = 10$ into their integrated expression: $10 = (2)^3 + 2(2)^2 + c$.
M1 for solving to find $c$: $10 = 8 + 8 + c \implies c = -6$.
A1 for the final equation $y = x^3 + 2x^2 - 6$.

Every question here has a full worked solution. See them all →

Question 3hard6 marks

Using the substitution $u^2 = 2x-1$, find $\int x\sqrt{2x-1} \, dx$.

Mark scheme: in the 7-day 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.

Revision notes: Integration · Integration techniques and areas · Differential equations

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In the practice engine

Skills you can practise

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.

Integrating powers of xDefinite integrals & area under a curveArea between a curve and a lineIntegrating e^x, 1/x and trig functionsIntegration by substitutionIntegration by partsIntegration using partial fractionsForming differential equationsSolving differential equations & modelling

Open Pure III · Calculus practice

Past papers

Past paper questions on integration

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 2025, June 2024, June 2023 Paper 2: June 2025, June 2024, June 2023
AQA 7357	Paper 1: June 2025, June 2024, June 2023 Paper 2: June 2025, June 2024, June 2023 Paper 3: June 2025, June 2024, June 2023
OCR H240	Paper 1: June 2025, June 2024, June 2023 Paper 2: June 2025, June 2024, June 2023 Paper 3: June 2025, June 2024, June 2023

Papers are © the exam boards; each page links to the official question paper and mark scheme. All A Level Maths past papers →

Integration: common questions

Is integration on the AQA, Edexcel and OCR A Level Maths exams?

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 Papers 1, 2 and 3; Edexcel Paper 1 and Paper 2; OCR A Paper 1, and the pure sections of Papers 2 and 3.

Is integration Year 12 or Year 13 content?

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.

When should I use integration by parts?

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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