plumiIntegration: OCR A‑Level Maths questions
Every OCR A-Level Maths exam question on P.8 Integration, 2018–2025, with its mark scheme and examiner’s report.
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![June 2025 Paper 1, question 12: 4 x + 12 (a) Express ( in partial fractions. [4] x 1 x )( 1 x ) + - d = y 2 4 x + The gradient of a curve is given by and the curve passes through the d x x ( 1](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24001-jun25-12-qp.webp?v=3)
![June 2025 Paper 2, question 8: In this question you must show detailed reasoning. 1 y Use integration by parts to find the exact value of 2 ln ( 3 2 x ) d x . [5] - 0](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24002-jun25-08-qp.webp?v=3)
![June 2025 Paper 3, question 5: (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 . =](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24003-jun25-05-qp.webp?v=3)
![June 2025 AS Paper 1, question 2: y 2 2 It is given that ( ax 2 x 4 )d x 45 , where a is a constant. - + = 0 Determine the value of a. [3]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h23001-jun25-02-qp.webp?v=3)
![June 2024 Paper 2, question 1: Differentiate the following with respect to x. 2 4 (a) 3 x [2] - 2 x (b) 4 x - 9 [2]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24002-jun24-01-qp.webp?v=3)
![June 2023 Paper 1, question 3: 3 (a) Given that ( ) f x x 2 x , use differentiation from first principles to show that f ( )x 2 x 2 . l = + = + [4] d = y (b) The gradient of a curve is given](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24001-jun23-03-qp.webp?v=3)
![June 2023 Paper 1, question 12: 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](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24001-jun23-12-qp.webp?v=3)
![June 2023 Paper 2, question 3: In this question you must show detailed reasoning. 1 Find the exact area of the region enclosed by the curve y + , the two axes and the = x 2 line x 2 5 . . [3]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24002-jun23-03-qp.webp?v=3)
All 58 questions on Integration, as a list
- June 2025, Paper 1, Q10 (12 marks): ex e x 2 1 10 The graph of y can be transformed to the graph of y - by a stretch parallel to the = = x-axis followed by a translation. (a) (i) State the scale f
- June 2025, Paper 1, Q12 (11 marks): 4 x + 12 (a) Express ( in partial fractions. [4] x 1 x )( 1 x ) + - d = y 2 4 x + The gradient of a curve is given by and the curve passes through the d x x ( 1
- June 2025, Paper 2, Q3 (7 marks): 3 The diagram shows part of the curve with equation y x + , between x = 0 and x = 2. 3 = y 3 2 1 0 x 0 1 2 (a) (i) Use the trapezium rule, with intervals of 0.5
- June 2025, Paper 2, Q8 (5 marks): In this question you must show detailed reasoning. 1 y Use integration by parts to find the exact value of 2 ln ( 3 2 x ) d x . [5] - 0
- 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 2025, AS Paper 1, Q2 (3 marks): y 2 2 It is given that ( ax 2 x 4 )d x 45 , where a is a constant. - + = 0 Determine the value of a. [3]
- June 2025, AS Paper 2, Q7 (9 marks): y (1, 0) a x O (3, –16) The diagram shows the curve with equation y f ( ) x where ( ) f x is a quadratic function. The = curve intersects the x-axis at the poin
- June 2024, Paper 1, Q1 (7 marks): y x O 2 - . x e x 2 The diagram shows part of the curve y = 2 y 2 – x (a) Use the trapezium rule with 4 intervals of equal width to find an estimate for x e d x
- 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 2024, Paper 2, Q1 (4 marks): Differentiate the following with respect to x. 2 4 (a) 3 x [2] - 2 x (b) 4 x - 9 [2]
- June 2024, Paper 2, Q4 (10 marks): - . 1 3 x 4 The diagram shows part of the graph of y xe = y 0.5 x 0 0.5 1 1.5 2 (a) Use the sign change method to determine, correct to 2 decimal places, the ro
- June 2024, Paper 2, Q5 (8 marks): A scientist is monitoring the decline in the population of a certain endangered species of animal in an area where their natural habitat has been damaged. As a
- June 2024, Paper 3, Q5 (13 marks): y x O M ^ h 3 2 y x x x The diagram shows the curve with equation 2 ln . The curve has a point of inflection = - at the point M. (a) (i) Show that the x-coordin
- June 2024, AS Paper 1, Q6 (7 marks): A curve has the following properties: d y • The gradient of the curve is given by d 2 x . =- x • The curve passes through the point ( , 4 13 ) . - Determine the
- June 2024, AS Paper 2, Q6 (7 marks): y x O 2 P 5 4 3 The diagram shows the curve with equation y x ax bx , where a and b are integers. The = + + curve has a minimum at the point P where x 2 . = The
- June 2023, Paper 1, Q3 (7 marks): 3 (a) Given that ( ) f x x 2 x , use differentiation from first principles to show that f ( )x 2 x 2 . l = + = + [4] d = y (b) The gradient of a curve is given
- 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 2023, Paper 2, Q3 (3 marks): In this question you must show detailed reasoning. 1 Find the exact area of the region enclosed by the curve y + , the two axes and the = x 2 line x 2 5 . . [3]
- June 2023, Paper 3, Q5 (9 marks): A mathematics department is designing a new emblem to place on the walls outside its classrooms. The design for the emblem is shown in the diagram below. y x O
- June 2023, Paper 3, Q7 (12 marks): 7 A car C is moving horizontally in a straight line with velocity ms v - at time t seconds, where - , of C is modelled by the equation 2 v 0 and t 0 . The accel
- June 2023, AS Paper 1, Q5 (6 marks): In this question you must show detailed reasoning. 3 The diagram shows part of the graph of y x 4 x . = - y x Determine the total area enclosed by the curve and
- June 2023, AS Paper 2, Q8 (7 marks): In this question you must show detailed reasoning. a 4 c 4 m y x Given that 3 d 7 , find the value of a. [7] + = x
- June 2022, Paper 1, Q1 (6 marks): y x O 2 The diagram shows part of the curve y x - . 1 = 3 y 2 (a) Use the trapezium rule with 4 intervals to find an estimate for x 1 d x . - 1 Give your answer
- 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
- June 2022, Paper 1, Q11 (9 marks): d y 2 3 x ln x 11 The gradient function of a curve is given by d . = x 3 y e The curve passes through the point (e, 1). 3 = y (a) Find the equation of this curv
- June 2022, Paper 2, Q3 (10 marks): 3 (a) Amaya and Ben integrated ( 1 )x , with respect to x, using different methods, as follows. + 3 x ( 1 ) + y 2 1 2 1 3 Amaya: ( 1 x ) d x c x x x c + = + = +
- June 2022, Paper 2, Q8 (7 marks): 50cm 30° hcm The diagram shows a water tank which is shaped as an inverted cone with semi-vertical angle 30° and height 50 cm. Initially the tank is full, and t
- June 2022, Paper 3, Q6 (8 marks): In this question you must show detailed reasoning. y y 2 x 9 = + x O – 2 x y 4 e – 1 = 2 x The diagram shows the curves y 2 x + and 9 y 4 e - 1 which intersect
- June 2022, AS Paper 1, Q5 (8 marks): d y 2 5 The gradient of a curve is given by d x 3 x . The curve passes through the point (6, 20). = - x (a) Determine the equation of the curve. [4] p 1y (b) He
- June 2022, AS Paper 2, Q8 (11 marks): + 8 (a) The quadratic polynomial ax bx , where a and b are constants, is denoted by ( ) f x . Use differentiation from first principles to determine, in terms o
- 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 2021, Paper 1, Q12 (13 marks): A cake is cooling so that, t minutes after it is removed from an oven, its temperature is i °C. When the cake is removed from the oven, its temperature is 160 °
- November 2021, Paper 2, Q6 (5 marks): x2 6 Alex is investigating the area, A, under the graph of y between x = and 1 x 1 5 . . They draw = = the graph, together with rectangles of width 0 1 . , and
- November 2021, Paper 3, Q7 (8 marks): A curve C in the x-y plane has the property that the gradient of the tangent at the point P x y is ( , ) three times the gradient of the line joining the point
- November 2021, Paper 3, Q8 (11 marks): y L R P M O x 2 x The diagram shows the curve M with equation y xe - . = (a) Show that M has a point of inflection at the point P where x = . 1 [5] The line L p
- November 2021, AS Paper 2, Q6 (6 marks): y d 2 6 A curve C has an equation which satisfies x 3 x + , for all values of x. 2 = 2 d (a) It is given that C has a single stationary point. Determine the nat
- November 2021, AS Paper 2, Q8 (11 marks): y l ,16 ^ h x O 6 The diagram shows the curve y 1 x and the line l, which is the normal to the curve at the = - + x point (1, 6). (a) Determine the equation of
- 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
- November 2020, Paper 1, Q12 (9 marks): Find the general solution of the differential equation d y 3 2 ( 2 x 3 x 11 x 6 ) y ( 20 x 35 ) . - - + = - d x Give your answer in the form y = f(x). [9]
- November 2020, Paper 2, Q8 (9 marks): d P 8 The rate of change of a certain population P at time t is modelled by the equation ( 100 P ) . = - d t Initially P = 2000. (a) Determine an expression for
- November 2020, Paper 3, Q5 (12 marks): y P R C x O 1 6 The diagram shows the curve C with parametric equations 3 - , where t 3 2 t x , y t e 2 0 . = = t The maximum point on C is denoted by P. (a) De
- November 2020, AS Paper 1, Q1 (6 marks): d c 5 m. 3 1 (a) Find x x 3 x [3] - + d 2 x c 2 d m y 2 (b) Find 6 x x . [3] - 3 x
- November 2020, AS Paper 1, Q5 (8 marks): The function f is defined by ( ) f x ( x a x )( 3 a x )( b ) where a and b are positive integers. = + + - (a) On the axes in the Printed Answer Booklet, sketch
- June 2019, Paper 1, Q8 (8 marks): A cylindrical tank is initially full of water. There is a small hole at the base of the tank out of which the water leaks. The height of water in the tank is x
- June 2019, Paper 1, Q11 (10 marks): y x O The diagram shows part of the curve y ln ( x 4 ) . = - y (a) Use integration by parts to show that ln ( x 4 ) d x ( x 4 ) ln x 4 x c . [5] - = - - - + (b)
- 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 2019, Paper 3, Q4 (14 marks): y P x O 1 r 2 1 The diagram shows the part of the curve y 3 x sin 2 x for which 0 x = G G . r 2 The maximum point on the curve is denoted by P. (a) Show that th
- June 2019, Paper 3, Q6 (10 marks): y x O 2 x 1 - The diagram shows part of the curve y . = 2 x x ( 2 3 )( 1 ) + + Find the exact area of the shaded region, giving your answer in the form p q ln r
- June 2019, AS Paper 1, Q1 (8 marks): 1 It is given that ( )x f 3 x . = - 3 x Find (a) f l ( )x , [3] (b) f ll ( )x , [2] y (c) f ( )x d x . [3]
- June 2019, AS Paper 2, Q7 (8 marks): y y ( 5 x x )( 1 ) = - - x a O y x x x a The diagram shows part of the curve ( 5 )( 1 ) and the line . = - - = Given that the total area of the regions shaded i
- June 2018, Paper 1, Q2 (4 marks): (i) Use the trapezium rule, with four strips each of width 0.5, to estimate the value of 2 y x 2 x e d 0 giving your answer correct to 3 significant figures. [3
- June 2018, Paper 1, Q12 (10 marks): In this question you must show detailed reasoning. y x 1 r ( , ) 0 4 4 cos 2 x 1 r The diagram shows the curve y , for x 0 , and the normal to the curve at the
- June 2018, Paper 1, Q13 (13 marks): A scientist is attempting to model the number of insects, N, present in a colony at time t weeks. When t = 0 there are 400 insects and when t = 1 there are 440
- June 2018, Paper 2, Q7 (7 marks): The diagram shows a part ABC of the curve y = 3 - 2x2, together with its reflections in the lines y = x, 7 y = -x and y = 0. y B A C x Find the area of the shad
- 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
- June 2018, Paper 3, Q7 (9 marks): = ^ h is given by the differential equation 7 y x The gradient of the curve f ^ h d y 3 2 2 x 1 4 y 0 - + = d x ^ h. By solving this differential equation show
- June 2018, AS Paper 1, Q4 (9 marks): 4 (i) It is given that y x 3 x . = + d . y (a) Find x [2] d (b) Find the values of x for which y is increasing. [2] ^ h y (ii) Find 3 4 x d x . [5] -
- June 2018, AS Paper 2, Q8 (9 marks): In this question you must show detailed reasoning. 7 1 The diagram shows part of the graph of y 2 x . The shaded region is enclosed by the curve, the x-axis = -
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