What is integration in the context of Cambridge International A Levels Pure Mathematics 2?

Integration is a fundamental concept in calculus, often described as the reverse process of differentiation. In the context of Cambridge International A Levels Pure Mathematics 2, integration is used to find the area under curves, solve differential equations, and calculate volumes of revolution. It involves finding an antiderivative or integral of a function, which represents the accumulation of quantities.

How do you perform integration by substitution in Pure Mathematics 2?

Integration by substitution is a method used to simplify the integration process by changing variables. In Pure Mathematics 2, you choose a substitution that simplifies the integrand, often by letting u be a function of x. Then, you express dx in terms of du and rewrite the integral in terms of u. After integrating with respect to u, you substitute back to express the result in terms of the original variable x.

What are definite and indefinite integrals, and how do they differ?

In Pure Mathematics 2, an indefinite integral represents a family of functions and includes a constant of integration, denoted as C. It is the antiderivative of a function. A definite integral, on the other hand, calculates the net area under a curve between two specific limits, providing a numerical value. The definite integral does not include a constant of integration and is evaluated using the Fundamental Theorem of Calculus.

How is integration used to find the area under a curve in A Level Mathematics?

To find the area under a curve using integration in A Level Mathematics, you set up a definite integral with the curve's equation as the integrand and the x-axis as the boundary. The limits of integration are the x-values that define the interval over which you want to find the area. By evaluating this definite integral, you obtain the total area between the curve and the x-axis over the specified interval.

What is the importance of the Fundamental Theorem of Calculus in integration?

The Fundamental Theorem of Calculus is crucial in integration as it links differentiation and integration, two core concepts in calculus. It states that if a function is continuous over an interval, then its integral can be computed using its antiderivative. This theorem provides a method to evaluate definite integrals by finding the difference between the values of an antiderivative at the upper and lower limits of integration, simplifying the calculation of areas and accumulated quantities.

Why is "Forgetting to divide by coefficient when integrating $e^{kx}$" a common mistake in Integration?

Why it happens: Treating as e^x. How to avoid it: e^kx dx = 1/ke^kx + c. The 1/k is essential. Check by differentiating. Source: 9709 Examiner Reports 2022-2024

Why is "Wrong sign on $\int \sin$" a common mistake in Integration?

Why it happens: Forgetting the minus. How to avoid it: x \, dx = - x + c. NEGATIVE. x \, dx = x + c (no minus). Source: 9709 Examiner Reports 2022-2024

Why is "Forgetting absolute value in $\ln|x|$" a common mistake in Integration?

Why it happens: Habit. How to avoid it: 1/x dx = |x| + c. Absolute value handles negative x. For definite integrals over positive interval, x is fine. Source: 9709 Examiner Reports 2022-2024

Pure Mathematics 2Cambridge International A Levels Mathematics (9709)

Integration

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Integration P2 Study Notes — Cambridge International A Level Mathematics 9709 (2024-2026 syllabus)

Integrating $e^{kx}$ , $\sin(kx)$ , $\cos(kx)$ , $1/x$ , $1/(ax+b)$ . The expanded P2 integration toolkit.

What you’ll learn

Mapped to the Cambridge IGCSE 9709 syllabus (2024-2026).

P2.5.1 — Integrate exponential and trig functions.
P2.5.2 — Integrate $1/x$ and $1/(ax+b)$ .
P2.5.3 — Apply to definite integrals.

Integrating $e^{kx}$ , $\sin(kx)$ , $\cos(kx)$

Reverse of derivatives.

Exponential. $\int e^{kx} \, dx = \frac{1}{k} e^{kx} + c$ .

Sine. $\int \sin(kx) \, dx = -\frac{1}{k}\cos(kx) + c$ . (Note MINUS.)

Cosine. $\int \cos(kx) \, dx = \frac{1}{k}\sin(kx) + c$ .

Coefficient $\frac{1}{k}$ . Required to "undo" the chain rule that would appear if we differentiated the result.

Example. $\int (3e^{2x} + 4x) \, dx$

$\int 3e^{2x} = \frac{3}{2}e^{2x}$ .
$\int 4x = 2x^2$ .
Total: $\frac{3}{2}e^{2x} + 2x^2 + c$ .

Example. $\int_0^{\pi/2} \cos 2x \, dx = \left[\frac{1}{2}\sin 2x\right]_0^{\pi/2} = \frac{1}{2}\sin\pi - 0 = 0$ .

Cambridge tip. Always check by differentiating your answer.

Coefficient $\frac{1}{k}$ always.
Note minus on $\sin$ .
Always verify via differentiation.

See the full worked example for integration →

Integrating $\frac{1}{x}$

$\ln|x| + c$ .

Standard. $\int \frac{1}{x} \, dx = \ln|x| + c$ .

Why $|x|$ ? $\ln$ is only defined for positive numbers, but $1/x$ is defined for $x < 0$ too. Absolute value handles both cases.

Linear denominator. $\int \frac{1}{ax + b} \, dx = \frac{1}{a}\ln|ax + b| + c$ .

General form. $\int \frac{f'(x)}{f(x)} \, dx = \ln|f(x)| + c$ .

Example. $\int_1^4 \frac{1}{x} \, dx = [\ln|x|]_1^4 = \ln 4 - \ln 1 = \ln 4$ .

Example. $\int \frac{1}{3x + 2} \, dx = \frac{1}{3} \ln|3x + 2| + c$ .

Cambridge tip. Use $\ln|x|$ unless told otherwise.

∫ 1/x dx = ln|x| + c, so the area under y = 1/x from 1 to a equals ln a — the defining property of the natural logarithm.

$\int \frac{1}{x} = \ln|x| + c$ .
Linear denom: divide by $a$ .
Recognise $f'/f$ pattern.

See the full worked example for integration →

Definite integrals and applications

Areas under exp/trig/log curves.

Areas. Apply $\int_a^b y \, dx$ with the new function families.

Volumes. Apply $V = \pi \int y^2 \, dx$ with squared expressions.

Solving differential equations. Simple separable: $\frac{dy}{dx} = f(x) g(y)$ . Separate: $\int \frac{dy}{g(y)} = \int f(x) \, dx$ . Integrate both sides.

Common patterns.

$\frac{dy}{dx} = ky$ → $y = Ce^{kx}$ (exponential growth/decay).
$\frac{dy}{dx} = k(A - y)$ → cooling-style equation.

Cambridge tip. For applications, sketch the region or curve when possible.

Areas and volumes with new functions.
Simple differential equations.
Exponential growth/decay model.

See the full worked example for integration →

How it’s examined

P2 integration appears in every paper. Most-tested: definite integral with $e^{kx}$ or trig (5-7 marks), area with log (6-8 marks), simple differential equation (8 marks).

Sources: Cambridge International A Level Mathematics 9709 syllabus (2024-2026); 9709 Examiner Reports 2022-2024; 9709/22 May/Jun 2024 question paper and mark scheme. Last reviewed 2026-05-11.

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Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.

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Step-by-step worked examples — Integration

Step-by-step solutions to past-paper-style questions on integration, written exactly the way a tutor would explain them at the board.

1Integration with exponential (5 marks)
Extended• Adapted from 9709/22 May/Jun 2024• exponential
Question
Find $\int (3e^{2x} + 4x) \, dx$ . (5 marks)
Step-by-step solution
1. Step 1
  $\int e^{kx} dx = \frac{1}{k} e^{kx}$ .
  $\int 3e^{2x} \, dx = \dfrac{3}{2} e^{2x}$
2. Step 2
  Power rule.
  $\int 4x \, dx = 2x^2$
3. Step 3
  Combine + constant.
  $\int (3e^{2x} + 4x) \, dx = \dfrac{3}{2}e^{2x} + 2x^2 + c$
Answer
$\frac{3}{2}e^{2x} + 2x^2 + c$ .
2Integration of trig (6 marks)
Extended• trig integration
Question
Evaluate $\int_0^{\pi/2} \cos 2x \, dx$ . (6 marks)
Step-by-step solution
1. Step 1
  $\int \cos(kx) dx = \frac{1}{k} \sin(kx)$ .
  $\int \cos 2x \, dx = \dfrac{1}{2} \sin 2x$
2. Step 2
  Apply limits.
  $\left[\dfrac{1}{2}\sin 2x\right]_0^{\pi/2} = \dfrac{1}{2}\sin\pi - \dfrac{1}{2}\sin 0 = 0 - 0 = 0$
Answer
$0$ .
3Integrate $1/x$ (5 marks)
Extended• log integration
Question
Find $\displaystyle\int_1^4 \dfrac{1}{x} \, dx$ . (5 marks)
Step-by-step solution
1. Step 1
  $\int \frac{1}{x} dx = \ln|x| + c$ .
2. Step 2
  Evaluate.
  $[\ln|x|]_1^4 = \ln 4 - \ln 1 = \ln 4$
Answer
$\ln 4 = 2\ln 2$ .

Key Formulae — Integration

The formulae you need to memorise for integration on the Cambridge International A Level 9709 paper, with every variable defined in plain English and a note on when to use it.

Exponential integral
$\displaystyle\int e^{kx} \, dx = \dfrac{1}{k} e^{kx} + c$
$k$
constant
When to use
Integrating exponential functions.
Trigonometric integrals
$\displaystyle\int \sin(kx) \, dx = -\dfrac{1}{k}\cos(kx) + c; \quad \int \cos(kx) \, dx = \dfrac{1}{k}\sin(kx) + c$
When to use
Integrating $\sin(kx)$ and $\cos(kx)$ .
$1/x$ integral
$\displaystyle\int \dfrac{1}{x} \, dx = \ln|x| + c$
When to use
Special case of power rule for $n = -1$ .
Generalised $1/x$
$\displaystyle\int \dfrac{1}{ax + b} \, dx = \dfrac{1}{a}\ln|ax + b| + c$
When to use
Integrating linear fractions.

Key Definitions and Keywords — Integration

Definitions to memorise and the exact keywords mark schemes credit for integration answers — sharpened from recent examiner reports for the 2026 Cambridge International A Level 9709 sitting.

Exponential integral
Examiner keyword
$\int e^{kx} \, dx = \frac{1}{k} e^{kx} + c$ . Reverse of derivative.
Log integral
Examiner keyword
$\int \frac{1}{x} \, dx = \ln|x| + c$ . The 'missing' power-rule case.
Trigonometric integral
Examiner keyword
$\int \sin x \, dx = -\cos x + c$ . $\int \cos x \, dx = \sin x + c$ .

Common Mistakes and Misconceptions — Integration

The traps other students keep falling into on integration questions — taken from recent Cambridge International A Level 9709 examiner reports and mark schemes — and how to avoid them.

✕Forgetting to divide by coefficient when integrating $e^{kx}$
9709 Examiner Reports 2022-2024
Why it happens
Treating as $e^x$ .
How to avoid it
$\int e^{kx} dx = \frac{1}{k}e^{kx} + c$ . The $\frac{1}{k}$ is essential. Check by differentiating.
✕Wrong sign on $\int \sin$
9709 Examiner Reports 2022-2024
Why it happens
Forgetting the minus.
How to avoid it
$\int \sin x \, dx = -\cos x + c$ . NEGATIVE. $\int \cos x \, dx = \sin x + c$ (no minus).
✕Forgetting absolute value in $\ln|x|$
9709 Examiner Reports 2022-2024
Why it happens
Habit.
How to avoid it
$\int \frac{1}{x} dx = \ln|x| + c$ . Absolute value handles negative $x$ . For definite integrals over positive interval, $\ln x$ is fine.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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Integration — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Integration

Short Study Notes in the form of Slides

Detailed Notes

What you’ll learn

How it’s examined

1Integration with exponential (5 marks)

2Integration of trig (6 marks)

3Integrate 1/x1/x1/x (5 marks)

Exponential integral

Trigonometric integrals

1/x1/x1/x integral

Generalised 1/x1/x1/x

Exponential integral

Log integral

Trigonometric integral

✕Forgetting to divide by coefficient when integrating ekxe^{kx}ekx

✕Wrong sign on ∫sin⁡\int \sin∫sin

✕Forgetting absolute value in ln⁡∣x∣\ln|x|ln∣x∣

Practice questions

Past paper style quiz

Assess your understanding

Integration — frequently asked questions

3Integrate $1/x$ (5 marks)

$1/x$ integral

Generalised $1/x$

✕Forgetting to divide by coefficient when integrating $e^{kx}$

✕Wrong sign on $\int \sin$

✕Forgetting absolute value in $\ln|x|$