plumiIntegration: WJEC A‑Level Maths questions
Every WJEC A-Level Maths exam question on P.8 Integration, 2018–2025, with its mark scheme and examiner’s report.
Find questions314
![Summer 2025 Unit 3, question 10: . for 0 X xX α, where 0 ~ α ~ 2 y sec x 10. (a) Sketch the graph of [1] sec d x x (b) Explain why . [2] 0](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/1300u30-jun25-10-qp.webp)
![Summer 2025 Unit 3, question 11: 5 2 x d x 11. Find , giving your answer in the form lna + b, where a, b are constants. 2 1 2 1 x x [10] 1](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/1300u30-jun25-11-qp.webp)
![Summer 2025 Unit 3, question 14: 2 x cos d x x 14. (a) Use integration by parts to find the exact value of . [6] 0 ln x d x u 1 ln x . (b) Use the substitution to find [5] 2 x 1 ln ](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/1300u30-jun25-14-qp.webp)
![Summer 2024 Unit 3, question 1: 1. The function f is given by 25 32 x . f x 2 5 1 2 x x x Express f x in terms of partial fractions. (a) [4] 2 f x d x ln P (b](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/1300u30-jun24-01-qp.webp)
![Summer 2024 Unit 3, question 6: Differentiate cosx from first principles. 6. (a) [5] Differentiate 3e sin4 x x with respect to x. (b) [3] 2sin2 d x x x (c) Find . [5]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/1300u30-jun24-06-qp.webp)
![Summer 2024 Unit 3, question 14: d y 1 lnx ln x y 14. (a) Given that , show that . [2] x d x x 2 (b) Hence, solve the differential equation 2 d x x t , d t ln x 1 t 3 x . given th](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/1300u30-jun24-14-qp.webp)
![Summer 2024 AS Unit 1, question 3: 5 3 2 2 d x x x 3. Find 4 . [3] ](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/2300u10-jun24-03-qp.webp)
![Summer 2023 Unit 3, question 8: 4 x 12 x 9 The function f is defined by 0 8 f x , where x 1 1. 2 2 x x – 3 5 ( ) can be written as 2 + Show that f x a) x −1 . [3] 7∫ ( ) b) Hence fi](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/1300u30-jun23-08-qp.webp)
![Summer 2023 Unit 3, question 12: 2 The rate of change of a variable y with respect to x is directly proportional to y. a) Write down a differential equation satisfied by y. [1] When x = 1 and y](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/1300u30-jun23-12-qp.webp)
![Summer 2023 Unit 3, question 14: 1∫ ( ) e2xdx . 3x −1 1 4 a) Use integration by parts to evaluate [4] 0 sin x d u 1 2cos x x b) Use the substitution to find . [4] 1 2cos x](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/1300u30-jun23-14-qp.webp)
![Summer 2023 AS Unit 1, question 13: ⎛ −2 ⎞ ∫ 3 + 5x3 + 7 1 3 a) Find 4x dx . [3] ⎝⎜ ⎠⎟ y x x 6 x 3 b) The diagram below shows the graph of . y x O Calculate the total area of the regi](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/2300u10-jun23-13-qp.webp)
![Summer 2022 Unit 3, question 5: 5 The function f (x) is defined by 6x + 4 ( ) = ) . f x ( ) x +1 ( ) 2x + 3 ( x −1 a) Express f (x) in terms of partial fractions. [4] ∫ 3x + 2 dx b) Find , giv](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/1300u30-jun22-05-qp.webp)
All 37 questions on Integration, as a list
- Summer 2025, Unit 3, Q4 (6 marks): 2 sec x x d 4. (a) Using the trapezium rule with 5 ordinates, find an approximate value for . 0 Show your working and give your answer correct to three decima
- Summer 2025, Unit 3, Q10 (3 marks): . for 0 X xX α, where 0 ~ α ~ 2 y sec x 10. (a) Sketch the graph of [1] sec d x x (b) Explain why . [2] 0
- Summer 2025, Unit 3, Q11 (10 marks): 5 2 x d x 11. Find , giving your answer in the form lna + b, where a, b are constants. 2 1 2 1 x x [10] 1
- Summer 2025, Unit 3, Q14 (11 marks): 2 x cos d x x 14. (a) Use integration by parts to find the exact value of . [6] 0 ln x d x u 1 ln x . (b) Use the substitution to find [5] 2 x 1 ln
- Summer 2025, Unit 4, Q7 (8 marks): A particle of mass 1.2 kg is moving in a straight line under the action of a variable force F N, 7. where 30 F . 2 t 2 The speed of the particle at time
- Summer 2025, AS Unit 1, Q4 (10 marks): y x 4 x 6 4. The diagram below shows a sketch of the curve C with equation and the line L y x+ 16 with equation . The curve and the line intersect at t
- Summer 2024, Unit 3, Q1 (11 marks): 1. The function f is given by 25 32 x . f x 2 5 1 2 x x x Express f x in terms of partial fractions. (a) [4] 2 f x d x ln P (b
- Summer 2024, Unit 3, Q6 (13 marks): Differentiate cosx from first principles. 6. (a) [5] Differentiate 3e sin4 x x with respect to x. (b) [3] 2sin2 d x x x (c) Find . [5]
- Summer 2024, Unit 3, Q9 (9 marks): y x x 1 9. The diagram below shows a sketch of the curve C1 with equation and a y cos2 x sketch of the curve C2 with equation . The curves intersect
- Summer 2024, Unit 3, Q14 (7 marks): d y 1 lnx ln x y 14. (a) Given that , show that . [2] x d x x 2 (b) Hence, solve the differential equation 2 d x x t , d t ln x 1 t 3 x . given th
- Summer 2024, Unit 4, Q9 (11 marks): 9. The diagram below shows a parcel, of mass m kg, sliding down a rough slope inclined at an 7 angle α to the horizontal, where sin . 25 α The coefficient o
- Summer 2024, AS Unit 1, Q3 (3 marks): 5 3 2 2 d x x x 3. Find 4 . [3]
- Summer 2024, AS Unit 1, Q14 (8 marks): y 2 3 x 2 x 14. The diagram below shows a sketch of the curve C with equation and the line y x 2 L with equation . The curve and the line intersect th
- Summer 2023, Unit 3, Q8 (7 marks): 4 x 12 x 9 The function f is defined by 0 8 f x , where x 1 1. 2 2 x x – 3 5 ( ) can be written as 2 + Show that f x a) x −1 . [3] 7∫ ( ) b) Hence fi
- Summer 2023, Unit 3, Q9 (8 marks): 9 The aerial view of a patio under construction is shown below. y 3 9x2 +16y2 = 144 x O 4 The curved edge of the patio is described by the equation 9x2 +16y2 =
- Summer 2023, Unit 3, Q12 (7 marks): 2 The rate of change of a variable y with respect to x is directly proportional to y. a) Write down a differential equation satisfied by y. [1] When x = 1 and y
- Summer 2023, Unit 3, Q14 (8 marks): 1∫ ( ) e2xdx . 3x −1 1 4 a) Use integration by parts to evaluate [4] 0 sin x d u 1 2cos x x b) Use the substitution to find . [4] 1 2cos x
- Summer 2023, Unit 4, Q10 (8 marks): A train is moving along a straight horizontal track. At time t seconds, its velocity is 1 0 v ms–1, its acceleration is a ms–2, and a is inversely proportional
- Summer 2023, AS Unit 1, Q13 (12 marks): ⎛ −2 ⎞ ∫ 3 + 5x3 + 7 1 3 a) Find 4x dx . [3] ⎝⎜ ⎠⎟ y x x 6 x 3 b) The diagram below shows the graph of . y x O Calculate the total area of the regi
- Summer 2022, Unit 3, Q5 (9 marks): 5 The function f (x) is defined by 6x + 4 ( ) = ) . f x ( ) x +1 ( ) 2x + 3 ( x −1 a) Express f (x) in terms of partial fractions. [4] ∫ 3x + 2 dx b) Find , giv
- Summer 2022, Unit 3, Q7 (7 marks): The diagram below shows a sketch of the curves y = x2 and . 0 7 y = 8 x y y = 8 x y = x2 x O Find the area of the region bounded by the two curves. [7]
- Summer 2022, Unit 3, Q14 (6 marks): π∫ x2 sinx dx . 1 4 Evaluate the integral [6] 0
- Summer 2022, Unit 3, Q18 (7 marks): 8 a) Use a suitable substitution to find ∫ x2 dx . [5] ( )4 x + 3 1∫ x2 b) Hence evaluate . dx [2] ( )4 x + 3 0
- Summer 2022, Unit 4, Q9 (8 marks): 9 Megan wants to make some ice cubes. She fills a plastic ice cube tray with water of temperature 10°C and places it directly into the freezer. The temperature
- Summer 2022, AS Unit 1, Q11 (15 marks): 1 The diagram below shows a sketch of the curve y = f (x), where f (x) = 10x + 3x2 − x3. The curve intersects the x-axis at the origin O and at the points A(–2,
- Summer 2019, Unit 3, Q1 (7 marks): 0 1 a) Express in terms of partial fractions. [4] ( ) x + 2 ( )2 x −1 ∫ 9 dx b) Find . [3] ( ) x + 2 ( )2 x −1
- Summer 2019, Unit 3, Q13 (8 marks): Wildflowers grow on the grass verge by the side of a motorway. The area populated by 1 3 wildflowers at time t years is A m2. The rate of increase of A is direc
- Summer 2019, Unit 3, Q14 (11 marks): ∫ Find (e2x + 6sin3x) dx. 1 4 a) [2] ∫ Find 7(x2 + sinx)6(2x + cosx) dx. b) [1] ∫ 1 c) Find . lnxdx [4] x2 Use the substitution u = 2cosx + 1 to evaluate d) π 3
- Summer 2019, Unit 4, Q8 (7 marks): 8 A box of mass 2 kg is projected along a horizontal surface with an initial velocity of 5 ms–1. The box experiences a variable resistive force of 0.4v2 N, wher
- Summer 2019, AS Unit 1, Q16 (8 marks): The diagram below shows a curve with equation y = (x + 2)(x – 2)(x + 1). 1 6 y x O Calculate the total area of the two shaded regions. [8]
- Summer 2018, Unit 3, Q5 (8 marks): 5 a) Show that 3x A B C , 2 ≡ ) + ) + ( ( x −1 x − 4 ( ) x − 4 ( ) ( ) x −1 x − 4 2 where A, B and C are constants to be found. [3] 3x 7∫ dx b) Evaluate , givin
- Summer 2018, Unit 3, Q14 (12 marks): 4 Evaluate 2∫ x3 ln x dx a) . [6] 1 2 + x 1∫ dx b) . [6] 4 − x2 0
- Summer 2018, Unit 3, Q15 (5 marks): 5 The variable y satisfies the differential equation 2 dy dx = 5 − 2y , where x x 0. Given that y = 1 when x = 0, find an expression for y in terms of x. [5]
- Summer 2018, Unit 4, Q7 (11 marks): An object of mass 0.5 kg is thrown vertically upwards with initial speed 24 ms–1. The 0 7 velocity of the object at time t seconds is v ms–1. During the upward
- Summer 2018, AS Unit 1, Q4 (6 marks): dy y = 5 x + 6 x 0 4 a) Given that , find when x = 8. [3] 3 dx (5x 2 +12x−5 + 7)dx 3 ∫ b) Find . [3]
- Summer 2018, AS Unit 1, Q6 (10 marks): 6 y L B C A x O The sketch shows the curve C with equation y = 14 + 5x – x2 and line L with equation y = x + 2. The line intersects the curve at the points A an
- Summer 2018, AS Unit 1, Q13 (11 marks): 3 A curve C has equation y = x3 – 3x2. a) Find the stationary points of C and determine their nature. [7] b) Draw a sketch of C, clearly indicating the stationa
Questions, mark schemes and examiner’s reports are © WJEC CBAC Ltd, from their published past papers. Plumi isn’t affiliated with any exam board. Want whole papers? Every WJEC A-Level Maths paper.
Rather write your answers by hand on iPad? Plumi marks them against the same mark schemes.