Use integration by parts to evaluate ∫01 xex dx.

[xex − ex]01 = (e − e) − (0 − 1) = 1.

Integration practice problems A level Maths, Topic 6

Written by STEM-Tutor's tutors · Updated October 2026 · 8 questions and 1 exam-style question

Integrating powers, e^x, 1/x and trigonometric functions, definite integrals and areas, substitution and integration by parts. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.

On this page

Key equations
Warm-up questions
Exam-style questions
Exam question practice
Common mistakes
Keep going

Key equations for integration

∫ xⁿ dx = xⁿ⁺¹/(n + 1) + c	n ≠ −1
∫ 1/x dx = ln\|x\| + c	the exception
∫ e^kx dx = e^kx/k + c	exponentials
∫ u dv = uv − ∫ v du	integration by parts

A formula booklet is provided in the exam. Answers within 2.5% are marked correct, and you can type fractions such as 1/6. What the booklet gives you, and what it leaves out, is in our A level Maths formula booklet guide.

Warm-up questions

Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.

Q1Warm-up

Evaluate ∫₀² 3x² dx.

Hint

Integrate, then substitute the limits.

Worked solution

[x³]₀² = 8 − 0.
= 8.

Q2Warm-up

Evaluate ∫₁⁴ 2x dx.

Hint

∫ 2x dx = x².

Worked solution

[x²]₁⁴ = 16 − 1.
= 15.

Q3Warm-up

What is ∫ 1/x dx?

Hint

The power rule fails for n = −1.

Worked solution

∫ 1/x dx = ln|x| + c: B.

Exam-style questions

Questions at the level of the A level Maths exam. The ones marked Challenge combine more than one idea.

Q4Exam-style

Find the area under y = x² + 1 between x = 0 and x = 3.

Hint

Integrate between the limits.

Worked solution

[x³/3 + x]₀³ = 9 + 3.
Area = 12.

Q5Exam-style

Evaluate ∫₀¹ e^2x dx, to 3 significant figures.

Hint

∫ e^2x dx = ½e^2x.

Worked solution

[½e^2x]₀¹ = ½(e² − 1).
= 3.19.

Q6Exam-style

Evaluate ∫₀^π/2 cos x dx.

Hint

∫ cos x dx = sin x.

Worked solution

[sin x]₀^π/2 = 1 − 0 = 1.

Q7Exam-styleYear 2

Use the substitution u = x² + 1 to evaluate ∫₀¹ 2x(x² + 1)³ dx.

Hint

du = 2x dx, and the limits become u = 1 to u = 2.

Worked solution

∫₁² u³ du = [u⁴/4]₁².
= (16 − 1)/4 = 3.75.

Q8ChallengeYear 2

Use integration by parts to evaluate ∫₀¹ xe^x dx.

Hint

Let u = x and dv = e^x dx.

Worked solution

∫ xe^x dx = xe^x − ∫ e^x dx = xe^x − e^x.
[xe^x − e^x]₀¹ = (e − e) − (0 − 1) = 1.

Exam question practice

One original exam-style question in the style of an A level paper. Answer it in full, with working, before you open the mark scheme: M marks are for method, A marks for accurate answers that follow from it, and B marks for correct statements on their own.

Exam-style question style · 8 points · about 8 minutes

The curve C has equation y = 4x − x². The line L has equation y = x.

(a) Find the x-coordinates where C meets the x-axis.
(b) Find the area enclosed by C and the x-axis.
(c) Find the area of the region enclosed by C and L.

Write your full answer on paper first, then open the scoring guide and tick each point you earned.

Scoring guide and model answer

Part (a)

4x − x² = x(4 − x) = 0, so x = 0 and x = 4.

1 point for: B1: x = 0 and x = 4

Part (b)

∫₀⁴ (4x − x²) dx = [2x² − x³/3]₀⁴ = 32 − 64/3 = 32/3.

1 point for: M1: integrating
1 point for: M1: substituting the limits
1 point for: A1: 32/3

Part (c)

4x − x² = x gives x = 0 or 3. ∫₀³ (3x − x²) dx = 27/2 − 9 = 4.5.

1 point for: M1: finding the intersection x = 3
1 point for: M1: integrating curve minus line
1 point for: M1: correct limits
1 point for: A1: 4.5

Your mark: 0 of 8

Common mistakes in integration

Forgetting + c on indefinite integrals.
Using x⁰/0 for ∫ 1/x dx instead of ln|x|.
Not changing the limits in a substitution, or changing them and then substituting back as well.
Taking line minus curve instead of upper minus lower, giving a negative area.

Keep going

Revise the ideas behind these questions:

Then try real exam questions: A level Maths past papers, with official scoring guidelines. · All A level Maths units