plumiIntegration: Edexcel International A Level Maths questions
Every Edexcel International A Level Maths exam question on P4.6 Integration, 2019–2025, with its mark scheme and examiner’s report.
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All 60 questions on Integration, as a list
- June 2025, Pure 4 (P4), Q3 (6 marks): Given that y = 4 at x = 6 π, solve the differential equation 2 d cos 2 3sin 2 0 d 4 4 y π π y x x y x x giving your answer in the form y 2 = g(x)
- June 2025, Pure 4 (P4), Q4 (8 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. f(x) = 5 17 10 1 2 1 2 x x x
- June 2025, Pure 4 (P4), Q7 (6 marks): (a) Show that the substitution u = tan x transforms tan tan ( sec ) x x x x 3 2 3 4 d to u k u u n ( ) 2 d where k and n are integers to be found. (b) Hence fin
- June 2025, Pure 4 (P4), Q9 (12 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. O 4 C x y 8 cm Figure 1 Figur
- June 2025 (A), Pure 4 (P4), Q1 (5 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Find, by algebraic integration, 2 3 1
- June 2025 (A), Pure 4 (P4), Q3 (5 marks): Given that y = 0 at x = 5 solve the differential equation 8 2 y y x x y d d e y 0 x 5 giving your answer in the form y 2 = f (x)
- June 2025 (A), Pure 4 (P4), Q7 (11 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. (a) Show that the substitution u = e x
- June 2025 (A), Pure 4 (P4), Q9 (7 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. O x y C R 20 a cm 20 cm 360π
- January 2025, Pure 4 (P4), Q1 (5 marks): y R O 8 x Figure 1 In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The curve
- January 2025, Pure 4 (P4), Q4 (10 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (i) The volume, V, of a spher
- January 2025, Pure 4 (P4), Q5 (10 marks): (i) Find x x x 2 4 ∫ e d writing the answer in simplest form. (ii) Use partial fractions and algebraic integration to show that 2 11 2 1 2 4 7 x x x x k d l
- January 2025, Pure 4 (P4), Q7 (6 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Use the substitution x = 4 si
- October 2024, Pure 4 (P4), Q6 (5 marks): Use the substitution u x 3 1 to show that 9 1 2 1 5 3 3 3 x x x x x A c k d where k and A are constants to be found and c is an arbitrary constant.
- October 2024, Pure 4 (P4), Q7 (8 marks): R 1 x y 4 O Figure 4 Figure 4 shows a sketch of part of the curve with equation y x x x 3 1 2 2 (a) Show that 3 1 2 2 x x A B x where A and B are constants to b
- October 2024, Pure 4 (P4), Q9 (10 marks): (a) Express 1 2 1 x x in partial fractions. The height above ground, h metres, of a carriage on a fairground ride is modelled by the differential equation d d h
- October 2024, Pure 4 (P4), Q10 (8 marks): R x y O Figure 5 Figure 5 shows a sketch of the curve with parametric equations 2 3 sin sin 2 0 2 π x t y t t t The region R, shown shaded in Figure 5, is bound
- June 2024, Pure 4 (P4), Q1 (5 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Find 6 0 cos3 d π x x x givin
- June 2024, Pure 4 (P4), Q5 (13 marks): t = a 1 t = b R x y O Figure 2 Figure 2 shows a sketch of the curve defined by the parametric equations x t t y t t a t b 2 2 2 3 ( ) where a and b are cons
- June 2024, Pure 4 (P4), Q7 (11 marks): The current, x amps, at time t seconds after a switch is closed in a particular electric circuit is modelled by the equation d d x t k x 3 where k is a constant
- June 2024, Pure 4 (P4), Q9 (9 marks): R 3 C x y O 1 3 Figure 3 The curve C, shown in Figure 3, has equation y x x x 1 4 1 arctan The region R, shown shaded in Figure 3, is bounded by C, the line w
- January 2024, Pure 4 (P4), Q2 (10 marks): Given that 3 4 2 2 1 2 1 2 2 1 2 2 x x x x A x B x C ( )( ) ( ) (a) find the values of the constants A, B and C. (b) Hence find the exact value of 3 4 2 2 1 2 7
- January 2024, Pure 4 (P4), Q5 (9 marks): (a) Find x x x 2 2 cos d ∫ (b) Hence solve the differential equation d d y t t t y cos 2 giving your answer in the form yn = f(t) where n is an integer.
- January 2024, Pure 4 (P4), Q7 (8 marks): (a) Using the substitution u = 4x + 2 sin 2x , or otherwise, show that e d e 4 2 2 2 2 0 2 1 8 1 x x x x sin cos π 2 y x O R Figure 3 The curve shown in
- October 2023, Pure 4 (P4), Q3 (12 marks): In this question you must show all stages of your working. Solutions based on calculator technology are not acceptable. (i) Use integration by parts to find the
- October 2023, Pure 4 (P4), Q7 (12 marks): The number of goats on an island is being monitored. When monitoring began there were 3000 goats on the island. In a simple model, the number of goats, x, in th
- October 2023, Pure 4 (P4), Q8 (14 marks): k O C x y 2 Figure 3 Figure 3 shows a sketch of the curve C with parametric equations 6 3sin 2 2cos 0 2 π x t t y t t The curve meets the y-axis at 2 and the x-
- June 2023, Pure 4 (P4), Q3 (11 marks): f ( ) x x x x x 8 5 2 1 4 3 1 > (a) Express f(x) in partial fractions. (b) Hence find f d ( ) x x ∫ (c) Use the answer to part (b) to find the value of k for wh
- June 2023, Pure 4 (P4), Q5 (10 marks): (i) Find x x x 2e d ∫ (ii) Use the substitution u = 1 3 −x to show that 27 1 3 2 1 3 1 2 x x x x Ax B k d ( ) ( ) where A and B are integers to be found and k i
- June 2023, Pure 4 (P4), Q6 (9 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The temperature, θ °C, of a c
- June 2023, Pure 4 (P4), Q8 (12 marks): y x R C P l Q O Figure 2 Figure 2 shows a sketch of part of the curve C with parametric equations x t t y t t t 1 1 0 7 > . The curve C intersects the x-axis at
- January 2023, Pure 4 (P4), Q3 (5 marks): O R 5 x y 5 Figure 1 In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 1
- January 2023, Pure 4 (P4), Q4 (9 marks): (a) Using the substitution u = 2 1 x + , show that 8 4 2 1 4 12 x x x + + ∫ e d may be expressed in the form ku u u a b 2e d ∫ where a, b and k are constants to
- January 2023, Pure 4 (P4), Q7 (12 marks): The volume V cm3 of a spherical balloon with radius r cm is given by the formula V = 4 3 πr3 (a) Find d d V r giving your answer in simplest form. At time t sec
- January 2023, Pure 4 (P4), Q8 (11 marks): O S l P C x y Figure 3 In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A curv
- October 2022, Pure 4 (P4), Q2 (7 marks): (a) Express 3 2 1 2 x x x − ( ) − ( ) in partial fraction form. (b) Hence show that 3 2 1 2 5 25 x x x x k − ( ) − ( ) = ∫ d ln where k is a fully simplified fr
- October 2022, Pure 4 (P4), Q5 (6 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. y x O k 1 2 R Figure 2 Figure
- October 2022, Pure 4 (P4), Q7 (12 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (i) Use the substitution u =
- October 2022, Pure 4 (P4), Q10 (8 marks): A spherical ball of ice of radius 12 cm is placed in a bucket of water. In a model of the situation, • the ball remains spherical as it melts • t minutes after
- June 2022, Pure 4 (P4), Q2 (9 marks): (a) Express 1 1 3 1 ( )( ) + − x x in partial fractions. (b) Hence find the solution of the differential equation (1 + 3x)(1 – x) d d y x = tan y – 1 3 < x 1
- June 2022, Pure 4 (P4), Q5 (8 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. (a) Use the substitution x = 2 sin u t
- June 2022, Pure 4 (P4), Q8 (10 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. R 20 cm O 10 x y Figure 2 Fig
- January 2022, Pure 4 (P4), Q5 (10 marks): y x t = 9 4 O R t = 0 Figure 2 Figure 2 shows a sketch of the curve with parametric equations x = 9 4 −t y = t t 3 9 4 + 0 t 9 4 The curve touches the x-axi
- January 2022, Pure 4 (P4), Q7 (8 marks): f (x) = 1 4 (4 − x)e x x 4 O 1 y 4 cm 2 cm Figure 3 Figure 4 Figure 3 shows the design of a doorknob. The shape of the doorknob is formed by rotating the curve
- January 2022, Pure 4 (P4), Q9 (11 marks): (a) Find the derivative with respect to y of 1 1 2 2 + ( ) ln y (b) Hence find a general solution to the differential equation 3cosec(2x) d d y x = y (1 + 2 ln
- October 2021, Pure 4 (P4), Q2 (6 marks): Find the particular solution of the differential equation d d y x = 4 4 5 2y x + x > −5 4 for which y = 1 3 at x = −1 4 giving your answer in the form y = f (x)
- October 2021, Pure 4 (P4), Q6 (7 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. O R x y 6 π 2 π Figure 2 Figure 2 show
- October 2021, Pure 4 (P4), Q8 (7 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. y = x ln x R e O x y Figure 3 (a) Find
- October 2021, Pure 4 (P4), Q9 (10 marks): 4.2 metres T h metres 8 metres Figure 4 Figure 4 shows a cylindrical tank that contains some water. The tank has an internal diameter of 8 m and an internal hei
- June 2021, Pure 4 (P4), Q2 (7 marks): Diagram NOT drawn to scale y x 9 R O 6 Figure 1 Figure 1 shows a sketch of part of the curve with equation y = 9 2 3 1 25 ( ) . x − x > 3 2 The finite region R,
- June 2021, Pure 4 (P4), Q4 (8 marks): Use algebraic integration and the substitution u = x to find the exact value of 10 5 2 1 4 x x x x + ∫ d Write your answer in the form 4ln a b , where
- June 2021, Pure 4 (P4), Q6 (10 marks): x O R y Figure 3 Figure 3 shows a sketch of the curve C with parametric equations x = 2 cos 2t y = 4 sin t 0 t 2 π The region R, shown shaded in Figure 3, i
- June 2021, Pure 4 (P4), Q8 (9 marks): (a) Given that y = 1 at x = 0, solve the differential equation d d y x = 6 1 3 2 xy x e y 0 giving your answer in the form y2 = g(x). (b) Hence find the equat
- January 2021, Pure 4 (P4), Q5 (8 marks): In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable. Using the substitution u x = + − 3 2
- January 2021, Pure 4 (P4), Q7 (7 marks): x O y R Figure 2 (a) Find e d 2x x x sin ∫ Figure 2 shows a sketch of part of the curve with equation y = e2x sin x x 0 The finite region R is bounded by the
- January 2021, Pure 4 (P4), Q9 (10 marks): x O y 3 Figure 3 Figure 3 shows a sketch of part of the curve with parametric equations tan 2sin 2 0 x θ y θ θ = = The finite region, shown shaded in Figure 3,
- January 2021, Pure 4 (P4), Q10 (14 marks): (a) Write 1 5 3 ( )( ) H H − + in partial fraction form. The depth of water in a storage tank is being monitored. The depth of water in the tank, H metres, is m
- October 2020, Pure 4 (P4), Q3 (6 marks): y x R O y = e 0.5x −2 Figure 1 Figure 1 shows a sketch of part of the curve with equation y = e 0.5x −2 The region R, shown shaded in Figure 1, is bounded by th
- October 2020, Pure 4 (P4), Q5 (7 marks): y O R 2 4 x Figure 3 (a) Find ln x x x 2 d ∫ Figure 3 shows a sketch of part of the curve with equation y x x x 3 2 2 ln x > 0.5 The finite region R, shown shad
- October 2020, Pure 4 (P4), Q7 (12 marks): (i) Using a suitable substitution, find, using calculus, the value of 3 2 1 1 5 x x x − ∫ d (Solutions relying entirely on calculator technology are not accepta
- October 2020, Pure 4 (P4), Q9 (9 marks): Bacteria are growing on the surface of a dish in a laboratory. The area of the dish, A cm2, covered by the bacteria, t days after the bacteria start to grow, is
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