Integration in 9709 P3: 10 Exam-Style Questions Worked

Integration in Pure Mathematics 3 is where Cambridge 9709 asks you to choose the right technique, not just apply one. In Paper 1 you integrated powers of (ax + b). In Paper 3 the toolkit widens: exponentials, logarithms, trigonometric functions, partial fractions, integration by parts and substitution, often combined in a single question.

This guide sets out what the syllabus requires, gives a simple decision process for choosing a method, and works through ten original exam-style questions, one for each core technique.

Where it sits

• Component: Paper 3, Pure Mathematics 3, syllabus section 3.5 (2026–2027 syllabus).
• Paper 3: 1 hour 50 minutes, 75 marks, 30% of the A Level. It is not part of the AS Level.
• Assumed knowledge: all of Pure Mathematics 1 integration, including definite integrals and areas.

What the syllabus requires

1. Integrate e^(ax+b), 1/(ax+b), sin(ax+b), cos(ax+b), sec²(ax+b) and 1/(x² + a²).
2. Use trigonometric relationships, such as double-angle formulae, to integrate (e.g. sin²x).
3. Integrate rational functions by partial fractions.
4. Recognise integrands of the form k f′(x)/f(x) and integrate them as logarithms.
5. Use integration by parts.
6. Use a given substitution to evaluate definite or indefinite integrals.

Choosing a method

Ask these questions in order: 1. Is it a standard form? (e^(ax+b), 1/(ax+b), sin, cos, sec², 1/(x² + a²)). Integrate directly. 2. Is the numerator a multiple of the derivative of the denominator? Use the logarithm rule. 3. Is it a rational function with a factorisable denominator? Use partial fractions. 4. Is it sin²x, cos²x or a similar power? Rewrite with a double-angle identity. 5. Is it a product of two different types of function (e.g. x·eˣ, x sin x, ln x)? Use integration by parts. 6. Is a substitution given? Use it. In 9709 the substitution is normally stated in the question.

Ten worked questions (original)

Q1. Exponential

Find ∫ e^(3x − 1) dx.

Divide by the coefficient of x: (1/3) e^(3x − 1) + c.

Q2. Logarithm from 1/(ax + b)

Find ∫ 1/(2x + 5) dx.

½ ln|2x + 5| + c. The ½ comes from the coefficient of x and is the most commonly omitted factor.

Q3. Double-angle identity

Find ∫ cos²x dx.

Using cos 2x = 2cos²x − 1, cos²x = ½(1 + cos 2x). ∫ ½(1 + cos 2x) dx = x/2 + (sin 2x)/4 + c.

Q4. sec² with a definite integral

Evaluate ∫ from 0 to π/8 of sec²(2x) dx.

= ½[tan 2x] from 0 to π/8 = ½(tan(π/4) − tan 0) = ½.

Q5. The inverse tangent form

Find ∫ 1/(x² + 9) dx.

With a = 3: (1/3) tan⁻¹(x/3) + c. This form is given in the MF19 list, but you must recognise when to use it.

Q6. Partial fractions

Evaluate ∫ from 2 to 3 of (5x + 1)/((x + 1)(x − 1)) dx, giving your answer as a single logarithm.

Let (5x + 1)/((x + 1)(x − 1)) = A/(x + 1) + B/(x − 1), so 5x + 1 = A(x − 1) + B(x + 1). - x = 1: 6 = 2B → B = 3 - x = −1: −4 = −2A → A = 2

∫ = [2 ln|x + 1| + 3 ln|x − 1|] from 2 to 3 = 2(ln 4 − ln 3) + 3(ln 2 − ln 1) = ln(16/9) + ln 8 = ln(128/9).

Q7. The f′(x)/f(x) form

Evaluate ∫ from 0 to 1 of x/(x² + 1) dx.

The derivative of x² + 1 is 2x, so the integrand is ½ × 2x/(x² + 1). = ½[ln(x² + 1)] from 0 to 1 = ½(ln 2 − ln 1) = ½ ln 2.

Q8. Integration by parts

Find ∫ x e^(2x) dx.

Let u = x, dv/dx = e^(2x), so du/dx = 1 and v = ½e^(2x). ∫ = ½x e^(2x) − ∫ ½e^(2x) dx = ½x e^(2x) − ¼e^(2x) + c.

Choosing u: pick the function that becomes simpler when differentiated (usually the polynomial), unless one factor is ln x. In that case u = ln x.

Q9. Parts with ln x

Evaluate ∫ from 1 to e of ln x dx.

Write ln x as 1 × ln x. Let u = ln x and dv/dx = 1, so du/dx = 1/x and v = x. ∫ = [x ln x] − ∫ 1 dx = [x ln x − x] from 1 to e = (e − e) − (0 − 1) = 1.

Q10. A given substitution

Use the substitution u = 1 + x² to evaluate ∫ from 0 to 1 of x(1 + x²)³ dx.

du/dx = 2x, so x dx = ½ du. Limits: x = 0 → u = 1; x = 1 → u = 2. ∫ = ½ ∫ from 1 to 2 of u³ du = ½[u⁴/4] from 1 to 2 = ½ × (16 − 1)/4 = 15/8.

Always change the limits when you substitute in a definite integral, or convert back to x before substituting the original limits. Mixing the two is a classic error.

When techniques combine

Paper 3 questions often chain two methods together. Typical patterns: - Partial fractions, then logarithms: as in Q6, the decomposition is worth marks in its own right, and the integration finishes the question. - Substitution, then parts: a given substitution may produce an integrand such as u eᵘ, which then needs integration by parts. - Integration inside a differential equation: separating variables (syllabus section 3.8) produces integrals that use exactly the techniques above. - Areas: a definite integral may be used to find the area of a region, so sketch the curve to check which parts lie below the axis.

Treat each stage as its own mini-question, and write a clear line of working between them.

Mistakes that cost marks

1. Missing coefficient factors with (ax + b): the 1/a is often forgotten.
2. Forgetting the modulus in ln|f(x)| when f(x) could be negative.
3. Sign errors with trigonometric integrals: ∫ sin x dx = −cos x + c.
4. Wrong choice of u in integration by parts, which makes the second integral harder than the first.
5. Not changing limits after substitution.
6. Losing exact form: if the question asks for an exact answer, leave it as ln(128/9) or ½ ln 2, not a decimal.

How to practise

• Use Kambridge Topicals to generate a 9709 Paper 3 integration PDF, with mark schemes interleaved.
• Then sit full Paper 3s from the 9709 past papers library under the 1 hour 50 minute limit.
• If Paper 1 calculus still feels shaky, revisit the 9709 Paper 1 guide first.

Frequently asked questions

Is the 1/(x² + a²) result given in the exam? It appears in the MF19 list of formulae, but you need to recognise the form and handle constants correctly.

Do I need to integrate tan x? It can be done with the f′(x)/f(x) method (tan x = sin x/cos x); the syllabus gives tan x as an example of this type.

Will the substitution always be given? The syllabus requires you to use a given substitution, so you will normally be told which one to use.

How much integration is on Paper 3? It varies by paper, but integration typically appears in several questions, often combined with differential equations or areas.

Master the techniques

Dr. Rizwan Jahangir, PhD in Mathematics (Sabancı University), teaches integration as a set of decisions, so you know which method to use and why.

Sources: Cambridge International AS & A Level Mathematics 9709 syllabus for examination in 2026–2027: section 3.5 and the Paper 3 assessment overview.

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