plumiIntegration by substitution: OCR A‑Level Maths questions
OCR A-Level Maths past-paper questions on 1.08.6 Integration by substitution, 2018–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.
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All 8 questions on Integration by substitution, as a list
- June 2025, Paper 3, Q5 (14 marks): (a) Use the substitution u cos x to show that = 1 cos x x + y d ln ( 1 cos x ) c . [4] = - + sin x Fig. 1 y x a O 1 cos x + Fig. 1 shows part of the curve y . =
- June 2024, Paper 1, Q12 (11 marks): In this question you must show detailed reasoning. y 2 x O 2 2 The diagram shows the curve with parametric equations x , y 2 t 3 t for t 0 . = = + H 4 t2 ( 1 )
- June 2023, Paper 1, Q12 (10 marks): x 12 (a) Use the substitution u e - to show that 2 = c c x 7 e 8 7 u 6 - + dd d x dd d u . = x 2 2 ( e 2 ) u ( u 2 ) - + [3] e e (b) Hence show that c ln 6 x 7
- June 2022, Paper 1, Q9 (7 marks): y 1 2 9 Use the substitution x 2 sin to show that 4 x d x . [7] = i - = r 3 1
- November 2021, Paper 1, Q11 (12 marks): 4 x y 4 2 2 2 2 11 (a) Use the substitution u x + to show that 3 d x ( x 6 ) x 3 c . [5] = = - + + 3 2 x 3 + (b) In this question you must show detailed reasoni
- November 2020, Paper 1, Q10 (11 marks): y x 1 2 O a k, for 1 The diagram shows the curve y sin x 1 1 x 2 . = - G G 2 a k 2 y 1 (a) Use rectangles of width 0.25 to find upper and lower bounds for sin x
- June 2019, Paper 2, Q1 (11 marks): (a) Differentiate the following. 2 x (i) + [3] x 2 1 2 - (ii) tan x ( 3 x ) [2] 1 (b) Use the substitution u x - to integrate 1 - . [4] = x 1 x 2 - (c) Integrat
- June 2018, Paper 3, Q5 (13 marks): (i) Use the trapezium rule, with two strips of equal width, to show that 4 = 1 11 x x [5] d 2 . . - 4 2 + 0 (ii) Use the substitution x = u2 to find the exact v
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