Mathematics 9709 · syllabus 1.8, 2.5, 3.5
Cambridge A Level Mathematics 9709 — Integration topical past papers
Integration as reverse differentiation, definite integrals, areas and volumes of revolution, the trapezium rule, and parts, substitution and partial fractions.
- Questions
- 368
- Years
- 2009–2026
- With mark scheme
- 99%
- Questions 2022–2026
- 110
Anyone can browse these pages. Opening the questions and their mark schemes takes a free Momentum account.
What Integration covers
In Paper 1 (section 1.8) integration is the reverse of differentiation. You integrate (ax + b)ⁿ for rational n other than −1, find a constant of integration from a point on a curve, and evaluate definite integrals, including simple improper ones. Definite integrals give the area between a curve and an axis, between a curve and a line or between two curves, and the volume formed when a region is rotated through 360° about one of the axes.
Paper 2 (section 2.5) adds the integrals of e^(ax + b), 1/(ax + b), sin(ax + b), cos(ax + b) and sec²(ax + b), the use of identities such as cos 2x ≡ 2cos²x − 1 to integrate sin²x and cos²x, and the trapezium rule, including whether it over- or under-estimates. Paper 3 (section 3.5) has all of that plus 1/(a² + x²), integrands of the form kf′(x)/f(x), partial fractions, integration by parts, and integration using a given substitution.
Every integration result is a derivative read backwards, so secure differentiation comes first, and Paper 3 integrals often need the partial fractions of algebra or a trigonometric identity before they can be done. Integration is in turn the tool for solving differential equations in Paper 3, for finding displacement from velocity in Mechanics, and for probabilities, means and variances of continuous random variables in Paper 6, which assumes the calculus of Paper 3.
In the syllabus
The headings this topic has in the Cambridge International 2026–2027 syllabus, as the board prints them.
- 1.8 Integration
- 2.5 Integration
- 3.5 Integration
What you should be able to do
- Integrate powers of x and of (ax + b), and find the constant of integration from a given point
- Evaluate definite integrals, including simple improper ones
- Find areas between curves, lines and axes, and volumes of revolution about either axis
- Integrate exponential, reciprocal and trigonometric functions of ax + b (Papers 2 and 3)
- Estimate an integral with the trapezium rule and say whether it over- or under-estimates (Papers 2 and 3)
- Integrate by parts, by a given substitution and by partial fractions (Paper 3)
Key ideas
- ∫(ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ / (a(n + 1)) + c, for n ≠ −1
- Volume of revolution: V = π∫y² dx about the x-axis, and V = π∫x² dy about the y-axis
- ∫e^(ax + b) dx = (1/a)e^(ax + b) + c and ∫1/(ax + b) dx = (1/a) ln|ax + b| + c
- Trapezium rule with strips of width h: ∫y dx ≈ ½h(y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ)
- Paper 3: ∫u (dv/dx) dx = uv − ∫v (du/dx) dx, and ∫f′(x)/f(x) dx = ln|f(x)| + c
How Integration has been examined
From 2022 to 2026, Integration was on Paper 1 · Pure Mathematics 1 in 11 of 11 exam series, worth about 20% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 1 · Pure Mathematics 1 | 59 | 20% | 11 of 11 |
| Paper 3 · Pure Mathematics 3 | 51 | 18% | 11 of 11 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 8.
Filed in Momentum under: Integration.
A worked example
The curve y = 6x − x² and the line y = 2x meet at the origin and at the point A. Find the coordinates of A and the area of the region enclosed between the curve and the line.
- Where they meet, 6x − x² = 2x, so x² − 4x = 0 and x = 0 or x = 4; A is (4, 8).
- Between x = 0 and x = 4 the curve lies above the line (at x = 1, 5 > 2), so the area is ∫₀⁴ ((6x − x²) − 2x) dx.
- Simplify the integrand to 4x − x² and integrate: [2x² − x³/3] from 0 to 4.
- Substitute the limits: (2 × 16 − 64/3) − 0 = 32 − 64/3 = 32/3.
Answer: A is (4, 8); the area is 32/3 square units (about 10.7)
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Forgetting to divide by a when integrating (ax + b)ⁿ: ∫(2x + 1)³ dx is (2x + 1)⁴/8 + c, not (2x + 1)⁴/4 + c.
- Finding an area with a single integral when the curve crosses the axis: the parts below the axis integrate to negative values and cancel, so each part must be integrated separately.
- Leaving out π, or squaring the wrong thing, in a volume of revolution: about the x-axis you integrate πy², with y written in terms of x.
- Changing the variable with a substitution but keeping the original limits: change the limits to the new variable, or substitute back before evaluating.
- Leaving out + c from an indefinite integral, or not using the given point to find it when the question asks for the equation of the curve.
Exam technique
- Sketch the region before finding an area or volume, so you know the limits and which curve is on top.
- Show the integrated expression and the substitution of both limits, and give exact answers, such as multiples of π or ln 2, when asked.
- In the trapezium rule, put the x- and y-values in a table, and find h from the number of strips, not the number of ordinates.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.