plumiIntegration: AQA A‑Level Maths questions
Every AQA A-Level Maths exam question on H Integration, 2018–2025, with its mark scheme and examiner’s report.
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![June 2025 Paper 1, question 5: Find ∫(4x – 3 + x – 1 2)dx [3 marks]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun25-05-qp.webp)
![June 2025 Paper 1, question 17: (a) Use the substitution u = ex + 1 to show that ∫ e2x ex + 1 dx = ex – ln(ex + 1) + k [5 marks] 17 (b) Solve the differential equation ( ex + 1 e2x ) dy dx = c](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun25-17-qp.webp)
![June 2025 Paper 3, question 8: (a) The expression x 2x2 + 3x + 1 can be written in the form A x + 1 + B 2x + 1 Find the value of A and the value of B [3 marks] 8 (b) Use your answer to part (](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73573-jun25-08-qp.webp)
![June 2024 Paper 3, question 6: (a) Find ∫ (6x2 – 5 √x) dx [3 marks] 6 (b) The gradient of a curve is given by dy dx = 6x2 – 5 √x The curve passes through the point (4, 90). Find the equation](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73573-jun24-06-qp.webp)
![June 2023 Paper 1, question 8: Show that (p 2 0 (x sin 4x) dx = − p 8 [6 marks]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun23-08-qp.webp)
![June 2023 Paper 1, question 14: (a) (i) Given that y = 2x write down dy dx [1 mark] 14 (a) (ii) Hence find ( 2x dx [2 marks] 14 (b) The area, A, bounded by the curve with equation y = 2x , the](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun23-14-qp.webp)
![June 2023 Paper 1, question 16: (a) Given that 1 16 − 9x2 × A 4 − 3x + B 4 + 3x find the values of A and B [3 marks] 16 (b) An empty container, in the shape of a cuboid, has length 1.6 metres,](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun23-16-qp.webp)
![June 2023 Paper 2, question 2: It is given that (6 0 f (x)dx = 20 and (6 3 f (x) dx = −10 Find the value of (3 0 f (x)dx Circle your answer. [1 mark] −30 −10 10 30](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73572-jun23-02-qp.webp)
![June 2023 Paper 3, question 8: Use the substitution u = x5 + 2 to show that (1 0 x9 (x5 + 2)3 dx = 1 180 [7 marks]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73573-jun23-08-qp.webp)
![June 2022 Paper 1, question 15: (a) Given that y = cosecy 15 (a) (i) Express y in terms of siny. [1 mark] 15 (a) (ii) Hence, prove that dy dy = × cosecy cot y [3 marks] 15 (a) (iii) Show that](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun22-15-qp.webp)
![June 2022 Paper 2, question 5: The binomial expansion of (2 + 5x)4 is given by (2 + 5x)4 = A + 160x + Bx2 + 1000x3 + 625x4 5 (a) Find the value of A and the value of B. [2 marks] 5 (b) Show t](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73572-jun22-05-qp.webp)
All 43 questions on Integration, as a list
- June 2025, Paper 1, Q5 (3 marks): Find ∫(4x – 3 + x – 1 2)dx [3 marks]
- June 2025, Paper 1, Q14 (8 marks): The graph of y = 4x sin 2x for 0 ≤ x ≤ π is shown below. x y O Show that the shaded area enclosed by the x-axis and the curve is kπ, where k is an integer to be
- June 2025, Paper 1, Q17 (10 marks): (a) Use the substitution u = ex + 1 to show that ∫ e2x ex + 1 dx = ex – ln(ex + 1) + k [5 marks] 17 (b) Solve the differential equation ( ex + 1 e2x ) dy dx = c
- June 2025, Paper 3, Q8 (7 marks): (a) The expression x 2x2 + 3x + 1 can be written in the form A x + 1 + B 2x + 1 Find the value of A and the value of B [3 marks] 8 (b) Use your answer to part (
- June 2024, Paper 1, Q18 (11 marks): (a) Use a suitable substitution to show that ∫ 4 0 (4x + 1)(2x + 1) 1 2 dx can be written as 1 2 ∫ 9 a (2u 3 2 – u 1 2)du where a is a constant to be found. [5
- June 2024, Paper 1, Q20 (10 marks): A gardener stores rainwater in a cylindrical container. The container has a height of 130 centimetres. The gardener empties the water from the container through
- June 2024, Paper 2, Q2 (1 mark): The graph of y = f (x) intersects the x-axis at (–3, 0), (0, 0) and (2, 0) as shown in the diagram below. y x A B 2 – 3 The shaded region A has an area of 189 T
- June 2024, Paper 3, Q6 (5 marks): (a) Find ∫ (6x2 – 5 √x) dx [3 marks] 6 (b) The gradient of a curve is given by dy dx = 6x2 – 5 √x The curve passes through the point (4, 90). Find the equation
- June 2024, Paper 3, Q11 (10 marks): The curve C with equation y = (x2 – 8x) In x is defined for x > 0 and is shown in the diagram below. The shaded region, R, lies below the x-axis and is bounded
- June 2023, Paper 1, Q8 (6 marks): Show that (p 2 0 (x sin 4x) dx = − p 8 [6 marks]
- June 2023, Paper 1, Q14 (11 marks): (a) (i) Given that y = 2x write down dy dx [1 mark] 14 (a) (ii) Hence find ( 2x dx [2 marks] 14 (b) The area, A, bounded by the curve with equation y = 2x , the
- June 2023, Paper 1, Q16 (14 marks): (a) Given that 1 16 − 9x2 × A 4 − 3x + B 4 + 3x find the values of A and B [3 marks] 16 (b) An empty container, in the shape of a cuboid, has length 1.6 metres,
- June 2023, Paper 2, Q2 (1 mark): It is given that (6 0 f (x)dx = 20 and (6 3 f (x) dx = −10 Find the value of (3 0 f (x)dx Circle your answer. [1 mark] −30 −10 10 30
- June 2023, Paper 2, Q14 (4 marks): A car has an initial velocity of 1 m s−1 The car is moving in a straight line. The acceleration a m s−2 of the car at time t seconds is given by a = 3kt2 − 2kt
- June 2023, Paper 3, Q8 (7 marks): Use the substitution u = x5 + 2 to show that (1 0 x9 (x5 + 2)3 dx = 1 180 [7 marks]
- June 2022, Paper 1, Q14 (9 marks): The region bounded by the curve y = (2x × 8) ln x and the x-axis is shaded in the diagram below. x 4 1 y 14 (a) Use the trapezium rule with 5 ordinates to find
- June 2022, Paper 1, Q15 (16 marks): (a) Given that y = cosecy 15 (a) (i) Express y in terms of siny. [1 mark] 15 (a) (ii) Hence, prove that dy dy = × cosecy cot y [3 marks] 15 (a) (iii) Show that
- June 2022, Paper 2, Q5 (6 marks): The binomial expansion of (2 + 5x)4 is given by (2 + 5x)4 = A + 160x + Bx2 + 1000x3 + 625x4 5 (a) Find the value of A and the value of B. [2 marks] 5 (b) Show t
- June 2022, Paper 2, Q10 (15 marks): A gardener has a greenhouse containing 900 tomato plants. The gardener notices that some of the tomato plants are damaged by insects. Initially there are 25 dam
- June 2022, Paper 3, Q2 (1 mark): The shaded region, shown in the diagram below, is defined by x2 − 7x + 7 × y × 7 − 2x x 5 O y Identify which of the following gives the area of the shaded regio
- June 2022, Paper 3, Q4 (2 marks): Find ( x2 + x 1 2 dx [2 marks]
- November 2021, Paper 1, Q10 (8 marks): (a) Given that y = tanx use the quotient rule to show that dy dx = sec2 x [3 marks] 10 (b) The region enclosed by the curve y = tan2 x and the horizontal line,
- November 2021, Paper 1, Q11 (8 marks): A curve, C, passes through the point with coordinates (1, 6) The gradient of C is given by dy dx = 1 6 (xy)2 Show that C intersects the coordinate axes at exact
- November 2021, Paper 1, Q14 (10 marks): The curve C is defined for t × 0 by the parametric equations x = t2 + t and y = 4t2 − t3 C is shown in the diagram below. x y O 14 (a) Find the gradient of C at
- November 2021, Paper 1, Q15 (10 marks): (a) Show that sin x − sinx cos 2x ≤ 2x3 for small values of x. [3 marks] 15 (b) Hence, show that the area between the graph with equation y = 8( sinx − sin x co
- November 2021, Paper 2, Q17 (11 marks): A ball is released from a great height so that it falls vertically downwards towards the surface of the Earth. 17 (a) Using a simple model, Andy predicts that t
- November 2021, Paper 3, Q8 (6 marks): Given that ( p 3 p 4 x cos x dx = ap + b find the exact value of a and the exact value of b. Fully justify your answer. [6 marks]
- November 2020, Paper 1, Q6 (4 marks): Four students, Tom, Josh, Floella and Georgia are attempting to complete the indefinite integral ( 1 x dx for x > 0 Each of the students’ solutions is shown bel
- November 2020, Paper 1, Q15 (10 marks): The region enclosed between the curves y = ex, y = 6 × e x 2 and the line x = 0 is shown shaded in the diagram below. y y = e x x O y = 6 – e x 2 Show that the
- November 2020, Paper 2, Q5 (6 marks): Use integration by substitution to show that (6 −1 4 x 4x + 1 p dx = 875 12 Fully justify your answer. [6 marks]
- November 2020, Paper 2, Q19 (8 marks): A particle moves so that its acceleration, a m s−2, at time t seconds may be modelled in terms of its velocity, v m s−1, as a = −0:1v2 The initial velocity of t
- November 2020, Paper 3, Q1 (1 mark): Given that (10 0 f (x)dx = 7 deduce the value of (10 0 f (x) + 1 dx Circle your answer. [1 mark] −3 7 8 17
- June 2019, Paper 1, Q14 (10 marks): The graph of y = 2x3 x2 + 1 is shown for 0 ≤ x ≤ 4 4 x 0 0 2 4 6 8 y 2 A Caroline is attempting to approximate the shaded area, A, under the curve using the tra
- June 2019, Paper 1, Q16 (16 marks): (a) y = e−x( sinx + cos x) Find dy dx Simplify your answer. [3 marks] 16 (b) Hence, show that ( e−x sinx dx = ae−x( sinx + cos x) + c where a is a rational numb
- June 2019, Paper 2, Q5 (7 marks): Solve the differential equation dt dx = ln x x2t for x > 0 given x = 1 when t = 2 Write your answer in the form t2 = f (x) [7 marks]
- June 2019, Paper 2, Q9 (9 marks): (a) Show that the first two terms of the binomial expansion of 4 − 2x2 p are 2 − x2 2 [2 marks] 9 (b) State the range of values of x for which the expansion fou
- June 2019, Paper 2, Q16 (16 marks): An elite athlete runs in a straight line to complete a 100-metre race. During the race, the athlete’s velocity, v m s−1, may be modelled by v = 11:71 − 11:68e−0
- June 2019, Paper 3, Q7 (8 marks): (a) Express 4x + 3 (x − 1)2 in the form A x − 1 + B (x − 1)2 [3 marks] 7 (b) Show that ( 4 3 4x + 3 (x − 1)2 dx = p + ln q where p and q are rational numbers. [
- June 2018, Paper 1, Q6 (12 marks): (a) Find the first three terms, in ascending powers of x, of the binomial expansion of 1 4 + x p [3 marks] 6 (b) Hence, find the first three terms of the binomi
- June 2018, Paper 2, Q3 (1 mark): The graph of y = x3 is shown. x 1 0 y 2 –1 –2 –3 3 4 Find the total shaded area. Circle your answer. [1 mark] −68 60 68 128
- June 2018, Paper 2, Q7 (8 marks): A function f has domain R and range fy 2 R : y × eg The graph of y = f (x) is shown. x y The gradient of the curve at the point (x, y) is given by dy dx = (x −
- June 2018, Paper 2, Q9 (14 marks): A market trader notices that daily sales are dependent on two variables: number of hours, t, after the stall opens total sales, x, in pounds since the stall ope
- June 2018, Paper 3, Q8 (9 marks): (a) Prove the identity sin 2x 1 + tan2 x × 2 sin x cos3 x [3 marks] 8 (b) Hence find ( 4 sin 4y 1 + tan2 2y dy [6 marks]
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