plumiIntegration: Edexcel International A Level Further Maths questions
Every Edexcel International A Level Further Maths exam question on FP3.4 Integration, 2019–2025, with its mark scheme and examiner’s report.
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All 35 questions on Integration, as a list
- June 2025, Further Pure 3 (FP3), Q2 (7 marks): Determine (i) 1 4 8 9 2 x x x d (ii) arcosh d 3x x ∫
- June 2025, Further Pure 3 (FP3), Q5 (9 marks): I x x x n n n 1 6 1 2 3 2 0 d (a) Show that, for n 1 3 6 4 8 6 2 1 n I nI n n n (b) Use the reduction formula in part (a) to determine the exact val
- June 2025, Further Pure 3 (FP3), Q7 (9 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. O y x Figure 1 Figure 1 shows
- January 2025, Further Pure 3 (FP3), Q3 (10 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (i) Determine 2 1 d 4 12 25 x
- January 2025, Further Pure 3 (FP3), Q5 (11 marks): y x C O Figure 1 In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 1 sho
- June 2024, Further Pure 3 (FP3), Q5 (8 marks): 4x 2 + 4x + 17 ≡ (2x + p) 2 + q where p and q are integers. (a) Determine the value of p and the value of q Given that 8 5 4 4 17 1 4 4 17 4 4 17 2 2 2 x x x x
- June 2024, Further Pure 3 (FP3), Q7 (8 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. y O x 1 2 Figure 1 Figure 1 shows the
- June 2024, Further Pure 3 (FP3), Q8 (9 marks): I x k x x n n k ( ) 1 2 0 d n 0 where k is a positive constant. (a) Show that I kn n I n n 2 3 2 1 n 1 Given that x k x x k 2 1 2 0 9 3 280 ( ) d (b) use th
- January 2024, Further Pure 3 (FP3), Q1 (7 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (i) Show that 4 3 2 4 8 d 16
- January 2024, Further Pure 3 (FP3), Q5 (11 marks): (a) Use the definitions of hyperbolic functions in terms of exponentials to prove that 1 – sech 2 x ≡ tanh 2 x I x x n n n n tanh ln 3 0 0 1 3 2 d (b) Show
- January 2024, Further Pure 3 (FP3), Q8 (9 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. O C R l x y Figure 2 Figure 2
- June 2023, Further Pure 3 (FP3), Q3 (11 marks): O x y Figure 1 Figure 1 shows a sketch of the curve C with equation 1 tan cot 2 6 3 π π y x x x (a) Show that the length of C is given by 3 2 2 6 1 tan cot d 2
- June 2023, Further Pure 3 (FP3), Q7 (9 marks): I x x n n n cosh 2 0 d (a) Show that, for n 2 I x x n n n I n n n cosh sinh 1 2 2 2 2 1 (b) Hence determine 1 2 3 cosh x x d collecting any like terms in yo
- June 2023, Further Pure 3 (FP3), Q8 (10 marks): (a) Differentiate x arcosh 5x with respect to x (b) Hence, or otherwise, show that arcosh d 5 3 20 2 2 5 2 1 4 1 4 3 5 x x p q r k ln ln where p, q, r and k
- January 2023, Further Pure 3 (FP3), Q4 (5 marks): (a) Determine 1 9 16 2 x x d (b) Hence determine the exact value of 1 9 16 2 2 2 x x d Give your answer in the form a b c ln( ), + 13 where a, b and c are rat
- January 2023, Further Pure 3 (FP3), Q6 (9 marks): A curve has parametric equations x a y a θ θ θ sin cos 1 where a is a positive constant. (a) Show that d d d d x y ka θ θ θ 2 2 2 2 2 sin where k is a constant
- January 2023, Further Pure 3 (FP3), Q8 (11 marks): I x x n n n cos d 0 (a) Prove that for I n x x n n I n n n 1 1 1 2 cos sin (b) Show that for positive even integers n cos ... ... n x x n n n n n d 0 2 1 3 5 3
- June 2022, Further Pure 3 (FP3), Q2 (9 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (i) Determine ∫ 1 5 4 2 + − x
- June 2022, Further Pure 3 (FP3), Q5 (7 marks): A curve has parametric equations x = 4e 1 2t y = et – t 0 t 4 The curve is rotated through 2π radians about the x‑axis. Show that the area of the curved sur
- June 2022, Further Pure 3 (FP3), Q7 (10 marks): In = ∫ x x x n 10 2 − ∫ d n ∈ ½x½ < 10 (a) Show that nIn = 10(n – 1)In – 2 – xn – 1(10 – x2) 1 2 n 2 (b) Hence find the exact value of x x x 5 2 10 − d ∫ 1
- January 2022, Further Pure 3 (FP3), Q2 (8 marks): O C x y Figure 1 Figure 1 shows a sketch of the curve C with parametric equations x = ln (sec θ + tan θ) – sin θ y = cos θ 0 θ 4 π The curve C is rotated th
- January 2022, Further Pure 3 (FP3), Q5 (7 marks): Determine (i) 1 3 5 2 x x x − + ∫ d (ii) 1 63 4 4 2 + − ∫ x x x d
- January 2022, Further Pure 3 (FP3), Q6 (10 marks): In = ∫e x sin n x dx n ∈ n 0 (a) Show that I x n x n x n n n I n x n n = + − + − + − − e sin (sin cos ) ( ) 1 2 2 2 1 1 1 n 2 (b) Hence find the exact val
- October 2021, Further Pure 3 (FP3), Q1 (6 marks): The curve C has equation y = 1 2 arcosh (2x) 7 2 x 13 Using calculus, determine the exact length of the curve C. Give your answer in the form p q , where p
- October 2021, Further Pure 3 (FP3), Q6 (9 marks): In = 2 0 π ∫ xn cos (x2) dx n 1 (a) Prove that, for n 5 In = 1 2 1 2 2 n π − − 1 4 (n − 1)(n − 3) In − 4 (b) Hence, determine the exact value of
- October 2021, Further Pure 3 (FP3), Q8 (13 marks): y = arccos (2 x ) (a) Determine d d y x (b) Show that y∫ dx = x arccos (2 x ) + x x 1 4 − ∫ dx (c) Use the substitution x = 1 2 cos θ to show that x x 1 4 0 1 8
- June 2021, Further Pure 3 (FP3), Q2 (7 marks): A curve has equation y = 9 2 −x 0 x 3 (a) Using calculus, show that the length of the curve is 3 2 π The curve is rotated through 2π radians about the x-axi
- June 2021, Further Pure 3 (FP3), Q5 (10 marks): In = ∫sec n x dx n 0 (a) Prove that for n 2 (n − 1)In = tan x sec n − 2 x + (n − 2)In − 2 (b) Hence, showing each step of your working, find the exact value
- June 2021, Further Pure 3 (FP3), Q7 (8 marks): Using calculus, find the exact values of (i) 1 4 5 2 1 2 x x x − + ∫ d (ii) x x x 2 2 3 3 3 − ∫ d
- January 2021, Further Pure 3 (FP3), Q4 (6 marks): Using the substitution x = 4 cosh θ show that 1 16 16 2 3 2 2 x x ax x c − ( ) = − + ∫ d | x | > 4 where a is a constant to be determined and c is an arbitrary
- January 2021, Further Pure 3 (FP3), Q6 (10 marks): I x x x n n = + ∫ 2 3 d n (a) Show that I x n x n n I n n n = + ( ) − − ( ) − − 1 2 1 2 2 3 3 1 n 3 (b) Hence show that x x x x x px q k 5 2 2 1 2 4 2 3 1
- January 2021, Further Pure 3 (FP3), Q8 (9 marks): The curve C has equation y = 2 + ln (1 − x2) 1 2 x 3 4 (a) Show that the length of the curve C is given by 1 1 2 2 1 2 3 4 + − ∫ x x x d (b) Hence
- October 2020, Further Pure 3 (FP3), Q2 (8 marks): Determine (i) 1 3 12 24 2x x x + + ∫ d (ii) 1 27 6 2 − − ∫ x x x d
- October 2020, Further Pure 3 (FP3), Q4 (9 marks): I x x x n n = ∫ cos d (a) Show that, for n 2 In = x n sin x + nx n–1 cos x – n(n – 1)In–2 (b) Hence find the functions f (x) and g (x) such that x x x x x x x
- October 2020, Further Pure 3 (FP3), Q7 (12 marks): The curve C has parametric equations x = cosh t + t, y = cosh t – t 0 t ln3 (a) Show that d d d d x t y t t + = 2 2 2 2cosh The curve C
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