plumiIntegration: Edexcel A‑Level Maths questions
Every Edexcel A-Level Maths exam question on P.8 Integration, 2019–2025, with its mark scheme and examiner’s report.
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All 54 questions on Integration, as a list
- June 2025, Pure 1, Q5 (3 marks): (a) Express 2.89 0 1.44 2 lim δx x δx x as an integral. (1) (b) Hence show that 2.89 0 1.44 2 lim δx x δx k x where k is an integer to be found. (Solutions rely
- June 2025, Pure 1, Q13 (5 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Show that x x x ( ) 2 1 2 25 3 0 2 d (
- June 2025, Pure 1, Q16 (7 marks): O 1 C R x y 3 Figure 5 Figure 5 shows a sketch of the curve C with equation y = 1 4 2 2 x x − 0 < x < 2 The region R, shown shaded in Figure 5, is bounded by C,
- June 2025, Pure 2, Q2 (4 marks): Find x x x 4 2 1 6 3 d giving the answer in simplest form. (4)
- June 2025, Pure 2, Q11 (11 marks): A company is trying to determine the most profitable selling price for a new toy. Given that • if the selling price of each toy is £30, the company expects to s
- June 2025, Pure 2, Q12 (8 marks): y x O Figure 3 Figure 3 shows a sketch of the graph with equation y = f (x) where f (x) = 4 ½x – 3½ – 5 x ∈ Given that a is a constant and ½a½ = 1 (a) find th
- June 2025, AS Pure, Q2 (5 marks): The line l passes through the points A (–3, 0) and B 5 2 22 , (a) Find the equation of l giving your answer in the form y = mx + c where m and c are constant
- June 2025, AS Pure, Q5 (6 marks): f (x) = x x x 5 2 12 4 − (a) Write f (x) in the form ax p + bx q where a, b, p and q are simplified constants. (3) (b) Hence find ∫f (x) dx giving your answer i
- June 2024, Pure 1, Q7 (8 marks): Diagram not drawn to scale. L 1.5 m H m Figure 2 Figure 2 shows a cylindrical tank of height 1.5 m. Initially the tank is full of water. The water starts to lea
- June 2024, Pure 1, Q10 (9 marks): A l1 R l2 O x y Figure 3 In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figu
- June 2024, Pure 1, Q13 (8 marks): (a) Given that a is a positive constant, use the substitution x = a sin 2 θ to show that 1 2 2 2 2 0 0 1 d sin 2 d 2 π a x a x x a θ θ (4) (b) Hence use algebra
- June 2024, Pure 1, Q14 (9 marks): A balloon is being inflated. In a simple model, • the balloon is modelled as a sphere • the rate of increase of the radius of the balloon is inversely proportio
- June 2024, Pure 2, Q11 (5 marks): O R C 1 x y Figure 5 In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 5
- June 2024, Pure 2, Q12 (12 marks): (a) Express 1 25 V V in partial fractions. (2) The volume, V microlitres, of a plant cell t hours after the plant is watered is modelled by the differential equ
- June 2024, AS Pure, Q1 (4 marks): Find ∫ 2 3 2 x x x − d giving your answer in simplest form. (4)
- June 2024, AS Pure, Q8 (10 marks): A B C R O y x l Figure 3 In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figu
- June 2023, Pure 1, Q1 (4 marks): Find x x x 1 2 2 5 3 ( ) d writing each term in simplest form. (4)
- June 2023, Pure 1, Q5 (6 marks): A continuous curve has equation y = f(x). The table shows corresponding values of x and y for this curve, where a and b are constants. x 3 3.2 3.4 3.6 3.8 4 y a
- June 2023, Pure 2, Q5 (5 marks): The curve C has equation y = f (x) The curve • passes through the point P(3, –10) • has a turning point at P Given that d d y x x x x k 2 9 5 3 2 where k is a c
- June 2023, Pure 2, Q10 (7 marks): f ( ) ( )( ) x kx x x 3 18 4 2 where k is a positive constant (a) Express f (x) in partial fractions in terms of k. (3) (b) Hence find the exact value of k for
- June 2023, Pure 2, Q11 (10 marks): h m 10 m 20 m 5 m Figure 1 A tank in the shape of a cuboid is being filled with water. The base of the tank measures 20 m by 10 m and the height of the tank is
- June 2023, Pure 2, Q13 (13 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Find the first three term
- June 2023, AS Pure, Q5 (5 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. O y 23 x y = 4x + 3 R 2 Figure 2 The f
- June 2023, AS Pure, Q16 (6 marks): A curve has equation y x x = f ( ), 0 Given that • f ( ) x x a x b 4 , where a and b are constants • the curve has a stationary point at 4 3 , • the curve meet
- June 2022, Pure 1, Q4 (3 marks): (a) Express δ 0 2.1 2 lim δ x x x x → =∑ as an integral. (1) (b) Hence show that 6.3 δ 0 2.1 2 lim δ x x x x → =∑ = ln k where k is a constant to be found. (2)
- June 2022, Pure 1, Q8 (8 marks): v t T O Figure 2 A car stops at two sets of traffic lights. Figure 2 shows a graph of the speed of the car, v ms−1, as it travels between the two sets of traffi
- June 2022, Pure 1, Q12 (5 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Show that x x x a b 3 8 1 2 ln d e e w
- June 2022, Pure 1, Q16 (9 marks): y 4 C R x O Figure 6 Figure 6 shows a sketch of the curve C with parametric equations x = 8 sin2 t y = 2 sin 2t + 3 sin t 0 t π 2 The region R, shown shaded
- June 2022, Pure 2, Q5 (6 marks): The table below shows corresponding values of x and y for y = log3 2x The values of y are given to 2 decimal places as appropriate. x 3 4.5 6 7.5 9 y 1.63 2 2.2
- June 2022, Pure 2, Q8 (6 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. y x O R Figure 3 Figure 3 shows a sket
- June 2022, Pure 2, Q14 (10 marks): (a) Express 3 2 1 1 ( )( ) x x − + in partial fractions. (3) When chemical A and chemical B are mixed, oxygen is produced. A scientist mixed these two chemicals
- June 2022, AS Pure, Q1 (4 marks): Find 8 3 2 5 3x x x − + ∫ d giving your answer in simplest form. (4)
- June 2022, AS Pure, Q10 (10 marks): P C O R y x l Figure 2 In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 2 shows
- November 2021, Pure 1, Q11 (8 marks): y R O x 2 4 Figure 2 Figure 2 shows a sketch of part of the curve with equation y = (ln x) 2 x > 0 The finite region R, shown shaded in Figure 2, is bounded by
- November 2021, Pure 2, Q12 (7 marks): (a) Use the substitution u = 1 + x to show that x x 1 0 16 + ∫ dx = 2 1 3 ( ) u u p q − ∫ du where p and q are constants to be found. (3) (b) Hence show that x
- November 2021, Pure 2, Q14 (12 marks): 8 m 3 m h m 5 m T Figure 5 Water flows at a constant rate into a large tank. The tank is a cuboid, with all sides of negligible thickness. The base of the tank
- November 2021, AS Pure, Q3 (4 marks): Find 3 4 2 4 3 x x x − ∫ d writing your answer in simplest form. (4)
- November 2021, AS Pure, Q9 (4 marks): Find the value of the constant k, 0 < k < 9, such that 6 20 9 x x k d = ∫ (4)
- November 2021, AS Pure, Q14 (10 marks): A curve C has equation y = f(x) where f(x) = –3x 2 + 12x + 8 (a) Write f(x) in the form a(x + b) 2 + c where a, b and c are constants to be found. (3) The curve
- November 2020, Pure 1, Q7 (5 marks): y x O l C R 25 (−2, 13) Figure 1 Figure 1 shows a sketch of a curve C with equation y = f (x) and a straight line l. The curve C meets l at the points (−2, 13)
- November 2020, Pure 1, Q10 (10 marks): (a) Use the substitution x = u2 + 1 to show that 3 1 3 2 1 6 3 2 5 10 d d x x x u u u p q − ( ) + − ( ) = + ( ) ∫ ∫ where p and q are positive constants to be f
- November 2020, Pure 1, Q14 (10 marks): A large spherical balloon is deflating. At time t seconds the balloon has radius r cm and volume V cm3 The volume of the balloon is modelled as decreasing at a
- November 2020, Pure 2, Q3 (5 marks): (a) Given that 2 log (4 − x) = log (x + 8) show that x2 − 9x + 8 = 0 (3) (b) (i) Write down the roots of the equation x2 − 9x + 8 = 0 (ii) State which of the ro
- November 2020, Pure 2, Q8 (6 marks): A curve C has equation y = f (x) Given that ● f ʹ (x) = 6x2 + ax − 23 where a is a constant ● the y intercept of C is −12 ● (x + 4) is a factor of f (x) find, i
- November 2020, Pure 2, Q12 (11 marks): y R O x Figure 3 The curve shown in Figure 3 has parametric equations x = 6 sin t y = 5 sin 2t 0 t 2 π The region R, shown shaded in Figure 3, is bounded by
- November 2020, AS Pure, Q7 (8 marks): Given that k is a positive constant and 5 2 3 4 1 x x k + = ∫ d (a) show that 3k + 5 k – 12 = 0 (4) (b) Hence, using algebra, find any values of k suc
- November 2020, AS Pure, Q10 (10 marks): g(x) = 2x3 + x2 – 41x – 70 (a) Use the factor theorem to show that g(x) is divisible by (x – 5). (2) (b) Hence, showing all your working, write g(x) as a produc
- June 2019, Pure 1, Q8 (10 marks): y O x R1 R2 b Figure 2 Figure 2 shows a sketch of part of the curve with equation y = x(x + 2)(x – 4). The region R1 shown shaded in Figure 2 is bounded by the
- June 2019, Pure 1, Q13 (11 marks): The curve C with equation y p x x q x = − − + 3 2 3 ( )( ) x , x x ≠− ≠ 3 2 , where S and q are constants, passes through the point 3 1 2 , ⎛ ⎝⎜ ⎞ ⎠⎟ and has
- June 2019, Pure 2, Q2 (4 marks): The speed of a small jet aircraft was measured every 5 seconds, starting from the time it turned onto a runway, until the time when it left the ground. The resu
- June 2019, Pure 2, Q5 (3 marks): y O x į[ y P 4 9 y x = Figure 3 Figure 3 shows a sketch of the curve with equation y x = The point P(x, y) lies on the curve. The rectangle, shown shaded on Fig
- June 2019, Pure 2, Q14 (15 marks): (a) Use the substitution u í h to show that dh h 4 − íOQ|í h |í h + k where k is a constant (6) A team of scientists is studying a species of
- June 2019, AS Pure, Q3 (6 marks): (a) Given that k is a constant, find 4 3 x kx + ⎛ ⎝⎜ ⎞ ⎠⎟ ∫ dx simplifying your answer. (3) (b) Hence find the value of k such that 4 3 0 5 2 x kx + ⎛ ⎝⎜ ⎞ ⎠⎟ ∫
- June 2019, AS Pure, Q13 (7 marks): y x O k R Figure 3 Figure 3 shows a sketch of part of the curve with equation y = 2x3 – 17x2 + 40x The curve has a minimum turning point at x = k. The region R,
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