AS & A Level Mathematics 9709 · Pure Mathematics 3 (for Paper 3) · 3.5

Integration: past paper questions

8 real questionsasked in 3 of the last 10 exam series2020–2025usually: Find, Show, Use

Real questions on integration

The newest 8 of 8 written questions. Open one to see it on the real paper.

9709/22 · October/November 2025 · Q5b[6]
6-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/31 · May/June 2025 · Q9a[5]
5-mark “Show” question — its maths and diagrams are best seen on the paper.
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9709/31 · May/June 2025 · Q9b[2]
2-mark question — its maths and diagrams are best seen on the paper.
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9709/31 · May/June 2025 · Q9c[3]
3-mark “Use” question — its maths and diagrams are best seen on the paper.
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9709/23 · May/June 2024 · Q5b[5]
5-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/22 · May/June 2021 · Q4a[3]
3-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/22 · May/June 2021 · Q4b[5]
5-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/21 · October/November 2020 · Q3[5]
5-mark “Find” question — its maths and diagrams are best seen on the paper.
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What examiners said

Many correct responses were seen, showing a good understanding of integration involving exponential functions. Some candidates did not gain the final accuracy mark if they had not given an exact answer in their solution, as requested by the question. This highlights the need to ensure that final answers are given in the required form.
— Examiner report, May/June 2021, 9709/22 Q4(a) (Cambridge)

It was essential that candidates realised that as an exact answer was required, they need to work with exact quantities rather than making use of their calculator. It was intended that the shaded area be found by calculating the area enclosed by the curve, the coordinate axes and the line x = 1, and then subtracting this from the area of the trapezium shown in the diagram. Attempts to find the equ...
— Examiner report, October/November 2020, 9709/21 Q3 (Cambridge)

Practise this topic

Answer these questions online and get each one marked against the official mark scheme, with what you missed and how to fix it.

Practise integration

In the syllabus

3.5 Integration

extend the idea of ‘reverse differentiation’ to
Including examples such as . 1 2 + 3x 2
ax + b
use trigonometrical relationships in carrying out e.g. use of double-angle formulae to integrate sin2 x
or cos2(2x).
integrate rational functions by means of Restricted to types of partial fractions as specified
in topic 3.1 above.
kf l^xh x
, tan x. f ^xh
, and e.g. integration of x +1
recognise when an integrand can usefully be e.g. integration of x sin 2x, x2e –x, ln x, x tan–1 x.
use a given substitution to simplify and evaluate e.g. to integrate sin2 2x cos x using the substitution

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