plumiIntegration: OCR MEI A‑Level Maths questions
Every OCR MEI 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 7: In this question you must show detailed reasoning. 2 Show that the area of the finite region enclosed by the curves y x x 7 + 2 = - 125 . 2 and y x x [7] 14 9 i](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64001-jun25-07-qp.webp?v=3)
![June 2025 Paper 1, question 10: x2 e 10 The diagram shows part of the graph of the function y + which is defined for x 2- . 1 = x 1 y 4 3 2 1 0 x –1 1 d . y (a) Find an expression for x [3] d](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64001-jun25-10-qp.webp?v=3)
![June 2025 Paper 2, question 14: 3 14 The equation of a curve is y + x . = 2 x (a) Determine the coordinates of the point where the curve cuts the x-axis. [2] d . y (b) (i) Find x [2] d 16 3 (i](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64002-jun25-14-qp.webp?v=3)
![June 2025 AS Paper 1, question 9: a k in the form x 1 3 5 p q 9 (a) Write the expression x x x [1] 3 . + + 2 2 a k. The curve passes through the point (1, –2). d y 1 3 5 (b) A curve has gradient](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h63001-jun25-09-qp.webp?v=3)
![June 2025 AS Paper 2, question 13: In this question you must show detailed reasoning Determine the exact value of c 6 2 d m 8 y x , + 3 2 )x x ( 1 giving your answer in its simplest form. [6]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h63002-jun25-13-qp.webp?v=3)
![June 2024 Paper 1, question 8: The equation of a curve is y sin 4 x 2 cos 2 x , where x is in radians. = + 4 2 (a) Show that, for small values of x, y 2 x 2 x . [2] . + - The diagram shows th](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64001-jun24-08-qp.webp?v=3)
![June 2023 Paper 1, question 3: ^ h y 2 4 - 3 Find x x x d x . [3]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64001-jun23-03-qp.webp?v=3)
![June 2023 Paper 2, question 10: ry 10 Determine the exact value of 4 x cos 2 d x x . [5] 4 0](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64002-jun23-10-qp.webp?v=3)
![June 2023 Paper 3, question 11: =/ 2 11 (a) Evaluate r . [1] r 1 (b) Show that Euler’s approximate formula, as given in line 13, gives the exact value of 5 =/ 2 r . [2] r 1](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64003-jun23-11-qp.webp?v=3)
![June 2023 Paper 3, question 12: With the aid of a suitable diagram, show that the three triangles referred to in line 26 have the areas given in line 27. [3]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64003-jun23-12-qp.webp?v=3)
![June 2023 Paper 3, question 13: n =/ 2 13 Prove that Euler’s approximate formula, as given in line 13, when applied to r gives exactly r 1 n n ( 1 2 )( n 1 ) + + . [4] 6 n](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64003-jun23-13-qp.webp?v=3)
![June 2023 Paper 3, question 14: =/ 1 13 6 n 5 + 14 Show that the expression given in line 33 simplifies to ln n , as given in . + + r 24 12 n n ( 1 ) + r 1 line 34. [3] 6](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64003-jun23-14-qp.webp?v=3)
All 57 questions on Integration, as a list
- June 2025, Paper 1, Q7 (7 marks): In this question you must show detailed reasoning. 2 Show that the area of the finite region enclosed by the curves y x x 7 + 2 = - 125 . 2 and y x x [7] 14 9 i
- June 2025, Paper 1, Q10 (10 marks): x2 e 10 The diagram shows part of the graph of the function y + which is defined for x 2- . 1 = x 1 y 4 3 2 1 0 x –1 1 d . y (a) Find an expression for x [3] d
- June 2025, Paper 2, Q14 (15 marks): 3 14 The equation of a curve is y + x . = 2 x (a) Determine the coordinates of the point where the curve cuts the x-axis. [2] d . y (b) (i) Find x [2] d 16 3 (i
- June 2025, Paper 2, Q16 (9 marks): In this question you must show detailed reasoning The population of puffins on a remote island is estimated to be P = 58.5, where P is measured in thousands of
- June 2025, AS Paper 1, Q9 (5 marks): a k in the form x 1 3 5 p q 9 (a) Write the expression x x x [1] 3 . + + 2 2 a k. The curve passes through the point (1, –2). d y 1 3 5 (b) A curve has gradient
- June 2025, AS Paper 2, Q13 (6 marks): In this question you must show detailed reasoning Determine the exact value of c 6 2 d m 8 y x , + 3 2 )x x ( 1 giving your answer in its simplest form. [6]
- June 2024, Paper 1, Q8 (6 marks): The equation of a curve is y sin 4 x 2 cos 2 x , where x is in radians. = + 4 2 (a) Show that, for small values of x, y 2 x 2 x . [2] . + - The diagram shows th
- June 2024, Paper 2, Q16 (12 marks): In this question you must show detailed reasoning. Find the particular solution of the differential equation d = y 9 y , d x ( x 1 )( x 2 ) - + given that x 2 w
- June 2024, Paper 3, Q3 (4 marks): In this question you must show detailed reasoning. x5 The diagram shows the curve with equation y and the square OABC where the points A, B = and C have coordin
- June 2024, Paper 3, Q5 (6 marks): In this question you must show detailed reasoning. Using the substitution u x + , find the value of the positive integer c such that 1 = x 1 c 4 +y d x ln 3 . [
- June 2024, AS Paper 1, Q12 (10 marks): The diagram shows the graph of ( ) f x k x ( p x )( q ) where k , p and q are constants. The graph = - - passes through the points (–1, 0), (0, –4) and (2, 0).
- June 2024, AS Paper 2, Q11 (6 marks): In this question you must show detailed reasoning. The equation of a curve is y 5 x - - for x x 6 0 . = H (a) Verify that the curve cuts the x-axis at x 4 and a
- June 2023, Paper 1, Q3 (3 marks): ^ h y 2 4 - 3 Find x x x d x . [3]
- June 2023, Paper 1, Q9 (10 marks): d y - . x x 9 The gradient of a curve is given by d e 4 e = - x (a) Show that the x-coordinate of any point on the curve at which the gradient is 3 satisfies th
- June 2023, Paper 2, Q10 (5 marks): ry 10 Determine the exact value of 4 x cos 2 d x x . [5] 4 0
- June 2023, Paper 2, Q11 (6 marks): In this question you must show detailed reasoning. d is directly proportional to the square root of x. y The variables x and y are such that x d d = . y When x
- June 2023, Paper 3, Q11 (3 marks): =/ 2 11 (a) Evaluate r . [1] r 1 (b) Show that Euler’s approximate formula, as given in line 13, gives the exact value of 5 =/ 2 r . [2] r 1
- June 2023, Paper 3, Q12 (3 marks): With the aid of a suitable diagram, show that the three triangles referred to in line 26 have the areas given in line 27. [3]
- June 2023, Paper 3, Q13 (4 marks): n =/ 2 13 Prove that Euler’s approximate formula, as given in line 13, when applied to r gives exactly r 1 n n ( 1 2 )( n 1 ) + + . [4] 6 n
- June 2023, Paper 3, Q14 (3 marks): =/ 1 13 6 n 5 + 14 Show that the expression given in line 33 simplifies to ln n , as given in . + + r 24 12 n n ( 1 ) + r 1 line 34. [3] 6
- June 2023, AS Paper 1, Q7 (6 marks): In this question you must show detailed reasoning. y x O Find the exact area of the shaded region shown in the diagram, enclosed by the x-axis and the 2 curve y
- June 2022, Paper 1, Q11 (6 marks): k = 2 11 Given that k is a positive constant, show that d x is inversely proportional to k. [6] 2 ( 2 x k ) + k
- June 2022, Paper 2, Q14 (8 marks): 14 Fig. 14.1 shows the curve with equation y , together with 5 rectangles of equal width. = 2 1 x + y 1.2 A B 1.0 C 0.8 D E 0.6 F 0.4 0.2 x 0 0.1 0.2 0.3 0.4 0.
- June 2022, Paper 3, Q6 (8 marks): A hot drink is cooling. The temperature of the drink at time t minutes is T °C. The rate of decrease in temperature of the drink is proportional to ( T 20 ) . -
- June 2022, AS Paper 1, Q6 (8 marks): d y 2 6 x x The curve passes through the The gradient of a curve is given by the equation 6 20 6 . = - + d x point (2, 6). (a) Find the equation of the curve. [
- June 2022, AS Paper 2, Q10 (9 marks): In this question you must show detailed reasoning. 3 2 The equation of a curve is y 12 x 24 x 60 x 72 . = - - + Determine the magnitude of the total area bounde
- November 2021, Paper 1, Q11 (11 marks): A balloon is being inflated. The balloon is modelled as a sphere with radius x cm at time t s. The 4 r volume V cm3 is given by V x3 . = 3 The rate of increase
- November 2021, Paper 2, Q16 (8 marks): In this question you must show detailed reasoning. x y Find d x . [8] x 1 +
- November 2021, Paper 3, Q7 (3 marks): y 7 Determine x cos 2 x x d . [3]
- November 2021, Paper 3, Q9 (11 marks): The diagram shows the curve y 3 x . = - y 5 4 3 2 1 x –1 0 1 2 3 4 5 6 7 8 –1 –2 (a) Draw the line y 5 x - on the copy of the diagram in the Printed Answer Book
- November 2021, Paper 3, Q10 (9 marks): 10 (a) Express ( in partial fractions. [3] 4 x 1 )( x 1 ) + + (b) A curve passes through the point (0, 2) and satisfies the differential equation d y y , = d x
- November 2021, AS Paper 1, Q7 (6 marks): The diagram shows part of a curve which passes through the point (1, 0). y x O 1 d y 8 The gradient of the curve is given by 6 x . = + d x 3 x Determine whether
- November 2021, AS Paper 2, Q4 (4 marks): J N 6 y 2 + KK OO 4 Find 9 x d x . [4] x L P
- November 2020, Paper 1, Q8 (7 marks): Fig. 8.1 shows the cross-section of a straight driveway 4 m wide made from tarmac. 4m Fig. 8.1 0 2 . The height h m of the cross-section at a displacement x m f
- November 2020, Paper 2, Q10 (9 marks): In this question you must show detailed reasoning. The equation of a curve is sin 2 x - . x y = x sin x (a) Use the small angle approximation given in the list
- November 2020, Paper 2, Q14 (8 marks): In this question you must show detailed reasoning. 1 - sin2x cosx. Fig. 14 shows the graphs of y = sin x cos 2x and y = 2 y 1 1 y= –sin2x cosx 2 0.5 x 0 0.5 1 y
- November 2020, Paper 3, Q7 (9 marks): 1 7 (a) Express x - as a single fraction. [1] + A x The population of fish in a lake is modelled by the differential equation d x x ( 400 x ) - d = t 400 where
- November 2020, Paper 3, Q8 (16 marks): 8 (a) The curve y is shown in Fig. 8. ^ h = 2 2 x 1 + y x O Fig. 8 2 2 d y 20 x 4 - (i) Show that . [5] = 2 2 4 d x ( 1 x ) + (ii) In this question you must sho
- November 2020, Paper 3, Q10 (2 marks): In this question you must show detailed reasoning. y r 1 Show that x x d ln 1 as given in line 37. [2] = r - e
- November 2020, AS Paper 2, Q6 (6 marks): Use integration to show that the area bounded by the x-axis and the curve with equation 4 square units. 2 y ( x 1 ) ( x 3 ) is 3 [6] = - -
- June 2019, Paper 1, Q1 (3 marks): In this question you must show detailed reasoning. ^ h 9 y 233 Show that 2 x x d x . [3] + = 3 4
- June 2019, Paper 1, Q15 (12 marks): A model for the motion of a small object falling through a thick fluid can be expressed using the differential equation d = v 9 8 . kv , - d t where v ms 1 - is
- June 2019, Paper 2, Q2 (4 marks): 12 2 Given that y ( x 5 ) , = + y d . (a) Find x [2] d ; 2 11 (b) Hence find 48 x x ( 5 ) d x . [2] +
- June 2019, Paper 2, Q4 (4 marks): 4 Fig. 4 shows the graph of y 1 x . = + y 1.5 1 0.5 x 0 –1 – 0.5 0.5 1 Fig. 4 -; 0 3 (a) Use the trapezium rule with h 0 5 . to find an estimate of 1 x d x , gi
- June 2019, Paper 2, Q6 (4 marks): < 32 6 x x [4] Find ln d . 5 x 6 Answer all the questions
- June 2019, Paper 3, Q3 (6 marks): 3 (a) Express ( in partial fractions. [3] x 2 )( x 3 ) + + < ^ h 1 (b) Find ( d x in the form ln f ( ) x c , where c is the constant of integration and ( ) f x
- June 2019, Paper 3, Q8 (10 marks): In this question you must show detailed reasoning. x3 A is the point (1, 0), B is the point (1, 1) and D is the point where the tangent to the curve y at = B cr
- June 2019, Paper 3, Q11 (6 marks): c x 11 By using the substitution dd [6] u 1 x , find d x . = + 1 x + e 8 Answer all the questions.
- June 2019, Paper 3, Q14 (4 marks): h / 2 2 14 (a) Express lim ( h y ) y as an integral. [1] - d y 0 " d 0 2 2 (b) Hence show that V r h , as given in line 41. [3] = r 3
- June 2019, Paper 3, Q15 (3 marks): A typical tube of toothpaste measures 5.4 cm across the straight edge at the top and is 12 cm high. It 3. contains 75 ml of toothpaste so it needs to have an in
- June 2019, AS Paper 2, Q7 (8 marks): c 11 d m y 3 7 (a) Find x x x [5] 15 . + 3 x c 11 m 8 y 3 11 (b) Show that x x x a , where a is a positive integer to be determined. [3] 15 d 2 + = # 3 x 0
- June 2018, Paper 1, Q8 (6 marks): 2 can be written as 8 (i) Show that sin 8 x cos x 1 cos x 4 . [3] - y 2 2 (ii) Hence find sin x cos x x d . [3]
- June 2018, Paper 1, Q10 (8 marks): Fig. 10 shows the graph of y ( k x ) ln x where k is a constant ( k 2 1 ) . = - y x O Fig. 10 Find, in terms of k, the area of the finite region between the cur
- June 2018, Paper 2, Q7 (4 marks): c 6 m y 7 x x [4] Find 4 d . - 3 x 5 Answer all the questions
- June 2018, Paper 2, Q17 (12 marks): ^ h in partial fractions. 2 (i) Express x 8 x 9 - + 17 [5] ^ ^h h 2 x 1 x 2 + - (ii) Express y in terms of x given that ^ h 2 d y y x 8 x 9 - + and y = 16 when
- June 2018, AS Paper 1, Q8 (8 marks): In this question you must show detailed reasoning. Fig. 8 shows the graph of a quadratic function. The graph crosses the axes at the points (-1, 0), (0, -4) and
- June 2018, AS Paper 2, Q6 (4 marks): y 6 x x [4] Show that ( 3 4 ) d 99 . + = 0
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