Unit 5: Calculus

Topic 5.16: Integration by substitution and by parts (HL only) Questions

Practice 20 exam-style questions for IB Math AA SL Topic 5.16. Review the question stems below, then unlock the full Question Bank to access markschemes, model answers, and AI grading.

11 mark

2026

For a definite integral, after substituting u you should:

Markscheme and model answer locked

Unlock Question

21 mark

2026

Integration by parts comes from reversing which rule?

Markscheme and model answer locked

Unlock Question

3Show that1 mark

2026

Show that

\int x e^{x} d x = e^{x} (x - 1) + c

by differentiating the right-hand side.

Markscheme and model answer locked

Unlock Question

41 mark

2026

In LIATE, which factor should be u in ∫ x³ ln x dx?

Markscheme and model answer locked

Unlock Question

51 mark

2026

To integrate ∫ 5(5x − 1)³ dx, the substitution u = 5x − 1 gives:

Markscheme and model answer locked

Unlock Question

61 mark

2026

After u = x² + 1 in ∫₀¹ 2x(x² + 1)⁵ dx, the integral becomes:

Markscheme and model answer locked

Unlock Question

7Find6 marks

2026

Use the substitution

u = x

to find

1 \int 4 \frac{1}{x + x} d x

. Give your answer correct to three significant figures.

Markscheme and model answer locked

Unlock Question

8Show that2 marks

2025

It is given that

\int x^{n} e^{x} d x = x^{n} e^{x} - n \int x^{n - 1} e^{x} d x

. Use this result to show that

\int x^{2} e^{x} d x = e^{x} (x^{2} - 2 x + 2) + c

Markscheme and model answer locked

Unlock Question

91 mark

2026

∫ x e^x dx equals:

Markscheme and model answer locked

Unlock Question

101 mark

2026

If after one round of parts the new integral looks HARDER, you should:

Markscheme and model answer locked

Unlock Question

11Show that4 marks

2025

A particle moves in a straight line with velocity

v (t) = t e^{- t /2}

m s

^{- 1}

for

t \geq 0

. Show that its displacement from

t = 0

t = a

4 - 2 (a + 2) e^{- a /2}

metres.

Markscheme and model answer locked

Unlock Question

121 mark

2026

How many applications of parts does ∫ x³ e^x dx need?

Markscheme and model answer locked

Unlock Question

13Show that2 marks

2025

Let

J = \int e^{x} cos x d x

. It is given that

J = e^{x} cos x + \int e^{x} sin x d x

and that

\int e^{x} sin x d x = e^{x} sin x - J

. Show that

J = \frac{1}{2} e^{x} (sin x + cos x) + c

Markscheme and model answer locked

Unlock Question

14Show that4 marks

2025

Let

m > 1

. Show that

1 \int m x^{2} ln x d x = \frac{m ^{3} ln m}{3} - \frac{m ^{3} - 1}{9}

Markscheme and model answer locked

Unlock Question

15Show that4 marks

2025

Show that

0 \int b x sin 2 x d x = \frac{sin 2 b - 2 b cos 2 b}{4}

Markscheme and model answer locked

Unlock Question

161 mark

2026

∫ 1/(2x + 5) dx equals:

Markscheme and model answer locked

Unlock Question

17Find5 marks

2025

Use integration by parts to find

0 \int 2 (x^{2} + 3) e^{- x} d x

. Give your answer correct to three significant figures.

Markscheme and model answer locked

Unlock Question

18Find3 marks

2026

Find ∫ x sin x dx.

Markscheme and model answer locked

Unlock Question

19Find3 marks

2026

Find ∫ sin⁴x cos x dx.

Markscheme and model answer locked

Unlock Question

201 mark

2026

Which integral is set up for u = cos x (du = −sin x dx)?

Markscheme and model answer locked

Unlock Question

Ready to practice Topic 5.16?

Get instant AI feedback on your answers, view detailed markschemes, and track your progress across all IB Math AA SL topics.

Start Practising Now Math AA SL Study Hub