A Level Maths: Pure 3 chapter 5: Integration MCQs with answers

By Team Megalecture · Updated 4 Nov 2026

27 exam-style questions on Integration, each with the answer and a short explanation. Ten of them are below to try. The chapter test draws ten fresh questions each time and is marked the moment you finish, with your score by topic.

27 questions · 1 topics · 10 per test

Test this chapter freeAll A Level Maths: Pure 3 chapters

What chapter 5 covers

Ten Integration questions with answers

Try each one, then open the answer. These ten stay the same; the test draws new ones every time.

1. Find the exact value of ∫26 6/(3x − 2) dx.

  1. A
    12 ln 2
  2. B
    4 ln 2
  3. C
    36 ln 2
  4. D
    2 ln 12
Show answer

Answer: B. ∫ 6/(3x − 2) dx = 6 × ⅓ ln(3x − 2) = 2 ln(3x − 2). So the value is 2 ln 16 − 2 ln 4 = 2 ln 4 = 4 ln 2. Forgetting to divide by the 3 gives 6 ln 4 = 12 ln 2; note that ln 16 − ln 4 = ln 4, not ln 12.

2. Find ∫ x2 ex dx.

  1. A
    ex(x2 + 2x + 2) + c
  2. B
    ex(x2 − 2x − 2) + c
  3. C
    ex(x2 − 2x + 2) + c
  4. D
    ex(x2 − 2x) + c
Show answer

Answer: C. Use parts twice: ∫ x2ex dx = x2ex − ∫ 2x ex dx, and ∫ 2x ex dx = 2x ex − 2ex. So the answer is ex(x2 − 2x + 2) + c. Stopping after one step, or losing the sign of the last term, gives the other forms.

3. To find ∫ x2 ln x dx by parts, using ∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx, which choice makes the new integral simpler, and why?

  1. A
    u = x2 and dv/dx = ln x, because ln x is a standard integral
  2. B
    u = ln x and dv/dx = x2, because ln x becomes 1/x when differentiated
  3. C
    u = x2 and dv/dx = ln x, because x2 becomes 2x when differentiated
  4. D
    u = ln x and dv/dx = x2, because x2 becomes 2x when differentiated
Show answer

Answer: B. With u = ln x, du/dx = 1/x and v = ⅓x3, so the new integrand is ⅓x2, a simple power. Choosing u = x2 needs v = ∫ ln x dx (itself a parts integral) and brings x2 ln x straight back. Here x2 is the part that is integrated, not differentiated.

4. Find the exact value of ∫03 1/(9 + x2) dx.

  1. A
    π/4
  2. B
    π/36
  3. C
    ln 2
  4. D
    π/12
Show answer

Answer: D. ∫ 1/(a2 + x2) dx = (1/a) tan−1(x/a) with a = 3, so the value is ⅓(tan−1 1 − tan−1 0) = ⅓ × ¼π = π/12. Leaving out the factor 1/a gives π/4; ln 2 comes from wrongly treating the integrand as f′(x)/f(x).

5. The substitution u = √x is used to find ∫19 1/(x + √x) dx. Which integral in terms of u is obtained?

  1. A
    ∫19 2/(u + 1) du
  2. B
    ∫13 1/(u2 + u) du
  3. C
    ∫13 1/(2u2(u + 1)) du
  4. D
    ∫13 2/(u + 1) du
Show answer

Answer: D. x = u2, so dx = 2u du, and x = 1, 9 give u = 1, 3. Then 1/(u2 + u) × 2u = 2/(u + 1). Forgetting to change the limits, or to replace dx by 2u du, are the usual slips.

6. Find the exact value of ∫0π/4 cos2 x dx.

  1. A
    π/8 + ¼
  2. B
    π/8 − ¼
  3. C
    π/4 + ½
  4. D
    π/8 + ½
Show answer

Answer: A. cos2 x = ½(1 + cos 2x), so the integral is [½x + ¼ sin 2x] from 0 to ¼π = π/8 + ¼. Forgetting the ½ in the identity doubles everything to π/4 + ½; integrating ½ cos 2x as ½ sin 2x gives π/8 + ½.

7. Find the exact value of ∫0π/12 sin 3x cos 3x dx.

  1. A
    1/12
  2. B
    1/6
  3. C
    ½
  4. D
    −1/12
Show answer

Answer: A. sin 3x cos 3x = ½ sin 6x, and ∫ ½ sin 6x dx = −(1/12) cos 6x. From 0 to π/12 this is −(1/12)(cos ½π − cos 0) = 1/12. Writing sin 3x cos 3x = sin 6x (dropping the ½) gives 1/6.

8. Find ∫ tan2 2x dx.

  1. A
    ½ tan 2x + x + c
  2. B
    2 tan 2x − x + c
  3. C
    ½ tan 2x − x + c
  4. D
    ⅙ tan3 2x + c
Show answer

Answer: C. Use tan2 θ = sec2 θ − 1: ∫ (sec2 2x − 1) dx = ½ tan 2x − x + c. Writing tan2 θ = sec2 θ + 1 gives the + x version.

9. Use the substitution u = 2x + 1 to find the exact value of ∫04 x/√(2x + 1) dx.

  1. A
    20/3
  2. B
    10/3
  3. C
    40/3
  4. D
    1/3
Show answer

Answer: B. x = ½(u − 1) and dx = ½ du, and the limits become u = 1 and u = 9. The integral is ¼∫19 (u1/2 − u−1/2) du = ¼[⅔u3/2 − 2u1/2] from 1 to 9 = ¼(12 + 4/3) = 10/3. Forgetting dx = ½ du gives 20/3; keeping the old limits 0 and 4 gives 1/3.

10. Find the exact value of ∫0π/6 sin 3x dx.

  1. A
    −⅓
  2. B
    3
  3. C
    1
  4. D
    ⅓
Show answer

Answer: D. ∫ sin 3x dx = −⅓ cos 3x. So the value is −⅓ cos ½π + ⅓ cos 0 = 0 + ⅓ = ⅓. Writing +⅓ cos 3x (a sign slip) gives −⅓; multiplying by 3 instead of dividing gives 3.

Start the chapter testAbout the A Level Maths: Pure 3 test

Read the notes for this chapter

Ways to practise this chapter

Integration: questions students ask

How many Integration MCQs are there?

27, each with the answer and a short explanation, across 1 topics: Integration. Every chapter test draws ten of them, so you can take it many times.

Is the A Level Maths: Pure 3 chapter 5 test free?

Yes. The test, the answers and the explanations are free and need no sign-up. Sign in only if you want your scores kept across devices.

How is the chapter test marked?

The moment you press Finish: your score, the right answer and an explanation for every question, and the ones you got wrong saved in My mistakes to retry.

Keep going

← Chapter 4: DifferentiationChapter 6: Numerical solution of equations →All chaptersStuck? Ask a question