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Integration by parts and by substitution

Past-paper questions pack

Name
Class
Date set
Date due
Score/ 102

Questions are numbered; the mark schemes (the answer key) follow on separate pages so the question pages can be handed out alone. www.mathsaleveltutor.com

Edexcel AS and A-level Maths past-paper questions since June 2018 · 21 parts, 102 marks · mark schemes follow the questions · www.mathsaleveltutor.com · Questions and mark schemes © Pearson Education Ltd

1. June 2025 · Paper 1 · Q13 · 5 marks
13. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Show that : x x d x ( ) (5)
2. June 2025 · Paper 1 · Q16(a) · 4 marks
16. y C R x Figure 5 Figure 5 shows a sketch of the curve C with equation y = 0 < x < 2 x - x The region R, shown shaded in Figure 5, is bounded by C, the line with equation x = 1, the x‑axis and the line with equation x = (a) Use the substitution x = 2 sin u to show that the area of R is given by b∫k cosec 2 u du a where a, b and k are constants to be found. (4)
3. June 2025 · Paper 2 · Q10(b) · 5 marks
3 of water in the container, Given that when the tap was opened, there was 0.25 m (b) solve the differential equation to show that –kt V = P – Qe where P, Q and k are positive constants to be found. (5)
4. June 2025 · Paper 2 · Q15(b) · 5 marks
(b) Using the substitution x = tan theta show that beta . / x x theta theta f d c os 2 d alpha where alpha and beta are constants to be found. (5)
5. June 2024 · Paper 1 · Q13(a) · 4 marks
13. (a) Given that a is a positive constant, use the substitution x = a sin 2 theta to show that pi a : : x a x x a theta theta d sin 2 d (4)
6. June 2024 · Paper 2 · Q11 · 5 marks
11. y C R x Figure 5 In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 5 shows a sketch of part of the curve C with equation x y x x <= e The finite region R, shown shaded in Figure 5, is bounded by • the curve C • the line with equation x = 1 • the x-axis Find the exact area of R, giving your answer in the form –3 A + Be where A and B are rational numbers to be found. (5)
7. June 2023 · Paper 2 · Q13(c) · 5 marks
(c) Find, using algebraic integration, the exact value of ∫ 0.4 x x d (3 + x ) 0.2 giving your answer in the form a ln b + c, where a, b and c are constants to be found. (5)
8. June 2023 · Paper 3 (Statistics) · Q6(c) · 4 marks
Xiang suggests that the frequency polygon based on this histogram could be modelled by a curve with equation y = kxe –x 0 x 4 where • x is measured in tens of hours • k is a constant (c) Use algebraic integration to show that n∫xe –x dx = 1 – (n + 1)e –n (4)
9. June 2022 · Paper 1 · Q12 · 5 marks
12. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Show that e : x x x a b ln d e where a and b are rational constants to be found. (5)
10. June 2022 · Paper 1 · Q16(b) · 4 marks
(b) Hence, using algebraic integration, find the exact area of R. (4)
11. Oct 2021 · Paper 1 · Q11(b) · 5 marks
(b) Use algebraic integration to find the exact area of R, giving your answer in the form y = a (ln 2) 2 + b ln 2 + c where a, b and c are integers to be found. (5)
12. Oct 2021 · Paper 2 · Q12(a) · 3 marks
12. (a) Use the substitution u = 1 + x to show that q ∫ ∫ x u - 1 3 ( ) dx = du u + x p where p and q are constants to be found. (3)
13. Oct 2021 · Paper 2 · Q12(b) · 4 marks
(b) Hence show that ∫ x dx = A – B ln 5 x + where A and B are constants to be found. (4)
14. Oct 2020 · Paper 1 · Q10(a) · 4 marks
10. (a) Use the substitution x = u2 + 1 to show that q ∫ ∫ x u d d = ( ) ( ) ( ) u + u x x - + - 1 3 p where p and q are positive constants to be found. (4)
15. Oct 2020 · Paper 2 · Q12(a)(ii) · 3 marks
(ii) Hence show, by algebraic integration, that the area of R is exactly 20 (3)
16. June 2019 · Paper 2 · Q14(a) · 6 marks
14. (a) Use the substitution u = 4 í h to show that dh œ = í8 ln௘|௘4 í h |௘í2 h + k 4 - h where k is a constant (6)
17. June 2019 · Paper 2 · Q14(c) · 7 marks
One of these trees is one metre high when it is first planted. According to the model, (c) calculate the time this tree would take to reach a height of 12 metres, giving your answer to 3 significant figures. (7)
18. June 2018 · Paper 1 · Q7(a) · 4 marks
7. Given that k + k ∫ (a) show that d is independent of k, x ( x - k ) (4) k
19. June 2018 · Paper 1 · Q7(b) · 3 marks
k ∫ (b) show that d is inversely proportional to k. x ( x - k ) (3) k
20. June 2018 · Paper 1 · Q13 · 7 marks
13. Show that ( ) ∫ x x + d x = 15 2 + (7)
21. June 2018 · Paper 2 · Q13 · 10 marks
13. y C l P(e, e) R x Figure 2 Figure 2 shows a sketch of part of the curve C with equation y = x ln x, x > 0 The line l is the normal to C at the point P(e, e) The region R, shown shaded in Figure 2, is bounded by the curve C, the line l and the x-axis. Show that the exact area of R is Ae2 + B where A and B are rational numbers to be found. (10)

Mark schemes

Each mark scheme shows the whole question; the parts on this topic are the ones listed above.

June 2025 · Paper 1 · Q13
Mark scheme June 2025 Paper 1 Q13
June 2025 · Paper 1 · Q16
Mark scheme June 2025 Paper 1 Q16(a)
June 2025 · Paper 2 · Q10
Mark scheme June 2025 Paper 2 Q10(b)
June 2025 · Paper 2 · Q15
Mark scheme June 2025 Paper 2 Q15(b)
June 2024 · Paper 1 · Q13
Mark scheme June 2024 Paper 1 Q13(a)
June 2024 · Paper 2 · Q11
Mark scheme June 2024 Paper 2 Q11
June 2023 · Paper 2 · Q13
Mark scheme June 2023 Paper 2 Q13(c)
June 2023 · Paper 3 (Statistics) · Q6
Mark scheme June 2023 Paper 3 (Statistics) Q6(c)
June 2022 · Paper 1 · Q12
Mark scheme June 2022 Paper 1 Q12
June 2022 · Paper 1 · Q16
Mark scheme June 2022 Paper 1 Q16(b)
Oct 2021 · Paper 1 · Q11
Mark scheme Oct 2021 Paper 1 Q11(b)
Oct 2021 · Paper 2 · Q12
Mark scheme Oct 2021 Paper 2 Q12(b)
Oct 2020 · Paper 1 · Q10
Mark scheme Oct 2020 Paper 1 Q10(a)
Oct 2020 · Paper 2 · Q12
Mark scheme Oct 2020 Paper 2 Q12(a)(ii)
June 2019 · Paper 2 · Q14
Mark scheme June 2019 Paper 2 Q14(c)
June 2018 · Paper 1 · Q7
Mark scheme June 2018 Paper 1 Q7(b)
June 2018 · Paper 1 · Q13
Mark scheme June 2018 Paper 1 Q13
June 2018 · Paper 2 · Q13
Mark scheme June 2018 Paper 2 Q13