AS & A Level Mathematics (9709)

Integration Banker

168 questions · about 8% of the marks we indexed · examined in 9 of 9 sessions.

Questions by year

Paper

2026

2026 May/June · Paper 11 · Q88 marksVolume of revolution
2026 May/June · Paper 11 · Q10 (b)5 marksCurve from gradient
2026 May/June · Paper 12 · Q57 marksCurve from gradient
2026 May/June · Paper 12 · Q9 (a)4 marksArea under curve
2026 May/June · Paper 13 · Q85 marksArea under curve
2026 May/June · Paper 15 · Q107 marksArea between curves
2026 May/June · Paper 21 · Q3 (a, b)4 marksArea / definite integrals
2026 May/June · Paper 21 · Q79 marksIntegration of trig functions
2026 May/June · Paper 22 · Q14 marksIntegration of exponentials
2026 May/June · Paper 22 · Q77 marksIntegration of trig functions
2026 May/June · Paper 23 · Q14 marksIntegration of exponentials
2026 May/June · Paper 23 · Q77 marksIntegration of trig functions
2026 May/June · Paper 25 · Q14 marksIntegration of exponentials
2026 May/June · Paper 25 · Q77 marksIntegration of trig functions
2026 May/June · Paper 31 · Q86 marksIntegration by parts
2026 May/June · Paper 32 · Q6 (b)6 marksIntegration by parts
2026 May/June · Paper 33 · Q66 marksIntegration by substitution
2026 May/June · Paper 35 · Q9 (b)4 marksIntegration of trig functions
2026 May/June · Paper 35 · Q10 (a, b)8 marksIntegration by substitution · Integration of trig functions

2025

2025 Oct/Nov · Paper 11 · Q8 (a, b)7 marksArea under curve · Volume of revolution
2025 Oct/Nov · Paper 11 · Q11 (c)5 marksCurve from gradient
2025 Oct/Nov · Paper 12 · Q4 (b)5 marksCurve from gradient
2025 Oct/Nov · Paper 12 · Q5 (b)5 marksArea under curve
2025 Oct/Nov · Paper 13 · Q98 marksVolume of revolution
2025 Oct/Nov · Paper 15 · Q8 (a, b)7 marksVolume of revolution
2025 Oct/Nov · Paper 15 · Q11 (b)6 marksArea between curves
2025 Oct/Nov · Paper 21 · Q13 marksIntegration of trig functions
2025 Oct/Nov · Paper 21 · Q5 (b)5 marksIntegration of exponentials
2025 Oct/Nov · Paper 22 · Q5 (b)6 marksIntegration of trig functions
2025 Oct/Nov · Paper 22 · Q6 (a)4 marksIntegration of exponentials
2025 Oct/Nov · Paper 23 · Q13 marksIntegration of trig functions
2025 Oct/Nov · Paper 23 · Q5 (b)5 marksIntegration of exponentials
2025 Oct/Nov · Paper 25 · Q13 marksIntegration of trig functions
2025 Oct/Nov · Paper 25 · Q5 (b)5 marksIntegration of exponentials
2025 Oct/Nov · Paper 31 · Q14 marksIntegration by parts
2025 Oct/Nov · Paper 32 · Q46 marksIntegration by parts
2025 Oct/Nov · Paper 32 · Q8 (b)5 marksIntegration of trig functions
2025 Oct/Nov · Paper 32 · Q9 (b)4 marksIntegration using partial fractions
2025 Oct/Nov · Paper 33 · Q66 marksIntegration by parts
2025 Oct/Nov · Paper 33 · Q11 (b)6 marksIntegration by substitution
2025 Oct/Nov · Paper 35 · Q7 (a, b)8 marksIntegration by substitution
2025 Oct/Nov · Paper 35 · Q10 (a)5 marksIntegration by parts
2025 May/June · Paper 11 · Q2 (b)4 marksCurve from gradient
2025 May/June · Paper 11 · Q45 marksArea between curves
2025 May/June · Paper 12 · Q66 marksArea between curves
2025 May/June · Paper 12 · Q9 (a, c)6 marksCurve from gradient
2025 May/June · Paper 13 · Q34 marksArea under curve
2025 May/June · Paper 13 · Q7 (b)4 marksCurve from gradient
2025 May/June · Paper 15 · Q14 marksCurve from gradient
2025 May/June · Paper 15 · Q45 marksVolume of revolution
2025 May/June · Paper 21 · Q4 (b)4 marksIntegration of exponentials
2025 May/June · Paper 21 · Q7 (c)4 marksIntegration of trig functions
2025 May/June · Paper 22 · Q13 marksIntegration (ln forms)
2025 May/June · Paper 22 · Q4 (c)4 marksArea / definite integrals
2025 May/June · Paper 23 · Q13 marksIntegration (ln forms)
2025 May/June · Paper 23 · Q4 (c)4 marksArea / definite integrals
2025 May/June · Paper 25 · Q13 marksIntegration (ln forms)
2025 May/June · Paper 25 · Q4 (c)4 marksArea / definite integrals
2025 May/June · Paper 31 · Q9 (a)5 marksIntegration by parts
2025 May/June · Paper 31 · Q11 (b)5 marksIntegration of trig functions
2025 May/June · Paper 32 · Q10 (b)6 marksIntegration by parts
2025 May/June · Paper 32 · Q11 (b)5 marksIntegration by substitution
2025 May/June · Paper 33 · Q34 marksIntegration of trig functions
2025 May/June · Paper 35 · Q7 (b)5 marksIntegration by parts

2024

2024 Oct/Nov · Paper 11 · Q5 (b)4 marksCurve from gradient
2024 Oct/Nov · Paper 11 · Q7 (b)4 marksArea under curve
2024 Oct/Nov · Paper 12 · Q7 (b)5 marksArea between curves
2024 Oct/Nov · Paper 12 · Q10 (b)4 marksCurve from gradient
2024 Oct/Nov · Paper 13 · Q9 (b)4 marksArea between curves
2024 Oct/Nov · Paper 21 · Q3 (b)4 marksIntegration of trig functions
2024 Oct/Nov · Paper 21 · Q5 (a)5 marksIntegration (ln forms)
2024 Oct/Nov · Paper 22 · Q2 (b)3 marksIntegration of trig functions
2024 Oct/Nov · Paper 22 · Q6 (a, b, c, d)9 marksArea / definite integrals · Integration (ln forms)
2024 Oct/Nov · Paper 23 · Q3 (b)4 marksIntegration of trig functions
2024 Oct/Nov · Paper 23 · Q5 (a)5 marksIntegration (ln forms)
2024 Oct/Nov · Paper 31 · Q25 marksIntegration by parts
2024 Oct/Nov · Paper 31 · Q6 (b)4 marksIntegration of trig functions
2024 Oct/Nov · Paper 32 · Q11 (b)9 marksIntegration by substitution
2024 Oct/Nov · Paper 33 · Q9 (b)5 marksIntegration (ln and tan^-1 forms)
2024 Oct/Nov · Paper 33 · Q11 (b)6 marksIntegration by substitution
2024 May/June · Paper 11 · Q96 marksOfficial paper ↗Volume of revolution
2024 May/June · Paper 12 · Q6 (b)5 marksArea between curves
2024 May/June · Paper 12 · Q9 (b)4 marksCurve from gradient
2024 May/June · Paper 13 · Q6 (a)4 marksCurve from gradient
2024 May/June · Paper 13 · Q9 (b)3 marksVolume of revolution
2024 May/June · Paper 21 · Q6 (a, b)9 marksOfficial paper ↗Area / definite integrals · Integration of trig functions
2024 May/June · Paper 21 · Q7 (b)6 marksOfficial paper ↗Integration (ln forms)
2024 May/June · Paper 22 · Q3 (b)5 marksIntegration of exponentials
2024 May/June · Paper 22 · Q5 (b)5 marksIntegration (ln forms)
2024 May/June · Paper 22 · Q7 (c)3 marksIntegration of trig functions
2024 May/June · Paper 23 · Q3 (b)5 marksIntegration of exponentials
2024 May/June · Paper 23 · Q5 (b)5 marksIntegration (ln forms)
2024 May/June · Paper 23 · Q7 (c)3 marksIntegration of trig functions
2024 May/June · Paper 31 · Q87 marksOfficial paper ↗Integration by substitution
2024 May/June · Paper 31 · Q10 (b)7 marksOfficial paper ↗Integration by parts
2024 May/June · Paper 32 · Q6 (b)5 marksIntegration by parts
2024 May/June · Paper 32 · Q7 (b)6 marksIntegration of trig functions
2024 May/June · Paper 33 · Q8 (b)5 marksIntegration of trig functions
2024 May/June · Paper 33 · Q119 marksIntegration by substitution

2023

2023 Oct/Nov · Paper 11 · Q8 (b)5 marksArea between curves
2023 Oct/Nov · Paper 11 · Q10 (a, b)6 marksCurve from gradient
2023 Oct/Nov · Paper 12 · Q3 (b)4 marksCurve from gradient
2023 Oct/Nov · Paper 12 · Q9 (b)5 marksArea between curves
2023 Oct/Nov · Paper 13 · Q14 marksCurve from gradient
2023 Oct/Nov · Paper 13 · Q11 (b)6 marksArea between curves
2023 Oct/Nov · Paper 21 · Q3 (a, b)6 marksIntegration (ln forms) · Integration of exponentials
2023 Oct/Nov · Paper 21 · Q6 (c)3 marksIntegration of trig functions
2023 Oct/Nov · Paper 22 · Q3 (b)3 marksArea / definite integrals
2023 Oct/Nov · Paper 22 · Q5 (b)5 marksIntegration (ln forms)
2023 Oct/Nov · Paper 22 · Q7 (c)5 marksIntegration of trig functions
2023 Oct/Nov · Paper 23 · Q3 (a, b)6 marksIntegration (ln forms) · Integration of exponentials
2023 Oct/Nov · Paper 23 · Q6 (c)3 marksIntegration of trig functions
2023 Oct/Nov · Paper 31 · Q9 (b)5 marksIntegration by substitution
2023 Oct/Nov · Paper 32 · Q56 marksIntegration (ln and tan^-1 forms)
2023 Oct/Nov · Paper 32 · Q9 (b)4 marksIntegration of trig functions
2023 Oct/Nov · Paper 33 · Q10 (b)5 marksIntegration by parts
2023 May/June · Paper 11 · Q10 (a)5 marksVolume of revolution
2023 May/June · Paper 11 · Q11 (c)3 marksCurve from gradient
2023 May/June · Paper 12 · Q13 marksCurve from gradient
2023 May/June · Paper 12 · Q54 marksArea under curve
2023 May/June · Paper 13 · Q9 (a)4 marksCurve from gradient
2023 May/June · Paper 13 · Q10 (c)5 marksArea between curves
2023 May/June · Paper 21 · Q3 (a)4 marksIntegration of exponentials
2023 May/June · Paper 21 · Q67 marksIntegration of trig functions
2023 May/June · Paper 22 · Q35 marksArea / definite integrals
2023 May/June · Paper 22 · Q6 (c)4 marksIntegration of trig functions
2023 May/June · Paper 23 · Q35 marksIntegration (ln forms)
2023 May/June · Paper 23 · Q6 (c)4 marksIntegration of trig functions
2023 May/June · Paper 31 · Q8 (b)5 marksIntegration using partial fractions
2023 May/June · Paper 31 · Q9 (a)5 marksIntegration by parts
2023 May/June · Paper 32 · Q9 (b)5 marksIntegration using partial fractions
2023 May/June · Paper 32 · Q10 (b)5 marksIntegration by substitution
2023 May/June · Paper 33 · Q7 (a, b)8 marksIntegration by parts · Integration by substitution

2022

2022 Oct/Nov · Paper 11 · Q2 (b)4 marksCurve from gradient
2022 Oct/Nov · Paper 11 · Q10 (a)5 marksArea between curves
2022 Oct/Nov · Paper 12 · Q8 (a)4 marksCurve from gradient
2022 Oct/Nov · Paper 12 · Q11 (b)5 marksArea between curves
2022 Oct/Nov · Paper 13 · Q7 (b)3 marksCurve from gradient
2022 Oct/Nov · Paper 13 · Q10 (b)4 marksVolume of revolution
2022 Oct/Nov · Paper 21 · Q6 (b)6 marksIntegration (ln forms)
2022 Oct/Nov · Paper 21 · Q7 (c)3 marksArea / definite integrals
2022 Oct/Nov · Paper 21 · Q8 (c)2 marksIntegration of trig functions
2022 Oct/Nov · Paper 22 · Q69 marksIntegration of exponentials
2022 Oct/Nov · Paper 23 · Q6 (b)6 marksIntegration (ln forms)
2022 Oct/Nov · Paper 23 · Q7 (c)3 marksArea / definite integrals
2022 Oct/Nov · Paper 23 · Q8 (c)2 marksIntegration of trig functions
2022 Oct/Nov · Paper 31 · Q9 (b)5 marksIntegration by parts
2022 Oct/Nov · Paper 32 · Q8 (a, b)8 marksIntegration by substitution
2022 Oct/Nov · Paper 32 · Q10 (b)5 marksIntegration using partial fractions
2022 Oct/Nov · Paper 33 · Q35 marksIntegration by parts
2022 Oct/Nov · Paper 33 · Q11 (b)5 marksIntegration using partial fractions
2022 May/June · Paper 11 · Q7 (b)5 marksArea between curves
2022 May/June · Paper 11 · Q10 (b)5 marksCurve from gradient
2022 May/June · Paper 12 · Q34 marksCurve from gradient
2022 May/June · Paper 12 · Q65 marksArea between curves
2022 May/June · Paper 13 · Q8 (b)4 marksArea between curves
2022 May/June · Paper 13 · Q10 (a)4 marksArea under curve
2022 May/June · Paper 21 · Q3 (a, b)7 marksIntegration of trig functions
2022 May/June · Paper 21 · Q7 (b)6 marksIntegration (ln forms)
2022 May/June · Paper 22 · Q4 (a, b)6 marksArea / definite integrals
2022 May/June · Paper 22 · Q6 (a)5 marksIntegration of exponentials
2022 May/June · Paper 22 · Q78 marksIntegration of trig functions
2022 May/June · Paper 23 · Q4 (a, b)6 marksArea / definite integrals
2022 May/June · Paper 23 · Q6 (a)5 marksIntegration of exponentials
2022 May/June · Paper 23 · Q78 marksIntegration of trig functions
2022 May/June · Paper 31 · Q6 (a, b)8 marksIntegration by substitution · Integration of trig functions
2022 May/June · Paper 32 · Q8 (b)5 marksIntegration using partial fractions
2022 May/June · Paper 33 · Q10 (a)5 marksIntegration by parts

References only, not question text. Open Official paper ↗ where Cambridge publishes the paper, or see the official Cambridge past-papers page; for other papers, ask your school. “Paper 42” means Paper 4, variant 2.

Practice questions

Original questions written by HomiLabs in the style of the exam, with answers and mark schemes. They are not Cambridge past paper questions and have not been reviewed by Cambridge. Try each one before opening the answer.

Q1Integration of trig functions · Standard · Paper 2 style · 6 marks

(a) Find ∫ (3 cos 2x − 4 sin(½x)) dx. [3]
(b) Hence find the exact value of ∫₀^{π/3} (3 cos 2x − 4 sin(½x)) dx. Give your answer in the form a√3 + b. [3]

Show answer

(a) (3/2) sin 2x + 8 cos(½x) + c
B1 (3/2) sin 2x; B1 +8 cos(½x) (coefficient 8 and sign); B1 + c.
(b) 19√3/4 − 8 (≈ 0.227)
M1 substitute both limits into the result of (a); sin(2π/3) = √3/2, cos(π/6) = √3/2, cos 0 = 1; (3√3/4 + 4√3) − 8 = 19√3/4 − 8 A1.

Q2Integration of trig functions · Standard · Paper 2 style · 4 marks

(a) Use the identity sec²x ≡ 1 + tan²x to express 2 tan²x + 1 in terms of sec²x. [1]
(b) Hence find the exact value of ∫₀^{π/4} (2 tan²x + 1) dx. [3]

Show answer

(a) 2 sec²x − 1
B1 2(sec²x − 1) + 1 = 2 sec²x − 1.
(b) 2 − π/4
M1 integrate to 2 tan x − x; M1 apply limits; A1 2(1) − π/4 = 2 − π/4.

Q3Integration of trig functions · Standard · Paper 2 style · 5 marks

(a) Express cos²3x in terms of cos 6x. [1]
(b) Hence show that ∫₀^{π/12} cos²3x dx = π/24 + 1/12. [4]

Show answer

(a) cos²3x = ½(1 + cos 6x)
B1 from cos 2A = 2cos²A − 1 with A = 3x.
(b) π/24 + 1/12 (≈ 0.214)
M1 integrate ½ + ½cos 6x; A1 x/2 + (1/12) sin 6x; M1 limits, sin(π/2) = 1; A1 π/24 + 1/12 shown.

Q4Integration of trig functions · Challenge · Paper 3 style · 6 marks

(a) By expanding sin(3x + x) and sin(3x − x), show that sin 3x cos x ≡ ½(sin 4x + sin 2x). [2]
(b) Hence find the exact value of ∫₀^{π/6} sin 3x cos x dx. [4]

Show answer

(a) Shown
M1 sin(3x+x) + sin(3x−x) = 2 sin 3x cos x; A1 rearrange to the given result.
(b) 5/16
M1 integrate to ½(−¼ cos 4x − ½ cos 2x); A1 correct; M1 limits, cos(2π/3) = −½, cos(π/3) = ½; A1 ½[(1/8 − 1/4) − (−3/4)] = ½ × 5/8 = 5/16.

Q5Integration of trig functions · Challenge · Paper 3 style · 6 marks

(a) Show that 1/(1 + cos 2x) ≡ ½ sec²x. [2]
(b) Hence find the exact value of ∫_{π/6}^{π/3} (1 + 2/(1 + cos 2x)) dx, giving your answer in the form aπ + b√3. [4]

Show answer

(a) Shown
M1 use 1 + cos 2x = 2cos²x; A1 1/(2cos²x) = ½ sec²x.
(b) π/6 + 2√3/3 (≈ 1.678)
M1 integrand = 1 + sec²x; A1 x + tan x; M1 limits: (π/3 + √3) − (π/6 + 1/√3); A1 π/6 + 2√3/3.

Q6Curve from gradient · Core · Paper 1 style · 4 marks

(a) A curve has gradient function dy/dx = 6x² − 4x + 1 and passes through the point (2, 9). Find the equation of the curve. [4]

Show answer

(a) y = 2x³ − 2x² + x − 1
M1 integrate (power of each term increased by 1); A1 2x³ − 2x² + x + c; M1 substitute (2, 9): 16 − 8 + 2 + c = 9; A1 c = −1.

Q7Curve from gradient · Standard · Paper 1 style · 8 marks

(a) A curve passes through the point (4, 5) and its gradient is given by dy/dx = 3/√(2x + 1). Find the equation of the curve. [4]
(b) Find the x-coordinate of the point where the curve crosses the x-axis. [2]
(c) Find the equation of the tangent to the curve at the point (4, 5). [2]

Show answer

(a) y = 3√(2x + 1) − 4
M1 integrate (2x+1)^(−½) to (2x+1)^(½) ÷ (½ × 2); A1 3(2x+1)^(½) + c; M1 substitute (4, 5): 9 + c = 5; A1 c = −4.
(b) x = 7/18
M1 3√(2x+1) = 4 so 2x + 1 = 16/9; A1 x = 7/18.
(c) y = x + 1
M1 gradient = 3/√9 = 1; A1 y − 5 = 1(x − 4), y = x + 1.

Q8Curve from gradient · Challenge · Paper 1 style · 8 marks

(a) A curve has d²y/dx² = 12x − 6. The curve has a stationary point at (1, 2). Find dy/dx and hence find the equation of the curve. [5]
(b) Find the coordinates of the other stationary point and determine the nature of each stationary point. [3]

Show answer

(a) dy/dx = 6x² − 6x; y = 2x³ − 3x² + 3
M1 integrate: dy/dx = 6x² − 6x + c; A1 dy/dx = 0 at x = 1 gives c = 0; M1 integrate again: y = 2x³ − 3x² + k; M1 substitute (1, 2); A1 k = 3.
(b) (0, 3) is a maximum; (1, 2) is a minimum
B1 6x² − 6x = 0 gives x = 0, y = 3; M1 d²y/dx² at x = 0 is −6 < 0, maximum; A1 at x = 1, d²y/dx² = 6 > 0, minimum.

Q9Integration by parts · Core · Paper 3 style · 4 marks

(a) Find the exact value of ∫₀¹ x e^{2x} dx. [4]

Show answer

(a) ¼(e² + 1) (≈ 2.097)
M1 parts with u = x, dv/dx = e^{2x}; A1 ½x e^{2x} − ¼e^{2x}; M1 apply limits; A1 (½e² − ¼e²) − (−¼) = ¼(e² + 1).

Q10Integration by parts · Standard · Paper 3 style · 5 marks

(a) Show that ∫₁^e x² ln x dx = (2e³ + 1)/9. [5]

Show answer

(a) (2e³ + 1)/9 (≈ 4.575)
M1 parts with u = ln x, dv/dx = x²; A1 (x³/3) ln x − ∫ x²/3 dx; A1 (x³/3) ln x − x³/9; M1 limits, ln e = 1, ln 1 = 0; A1 e³/3 − e³/9 + 1/9 = (2e³ + 1)/9.

Q11Integration by parts · Challenge · Paper 3 style · 6 marks

(a) Use integration by parts twice to find the exact value of ∫₀^{π/2} x² cos x dx. [6]

Show answer

(a) π²/4 − 2 (≈ 0.467)
M1 first parts: x² sin x − ∫ 2x sin x dx; A1; M1 second parts: ∫ 2x sin x dx = −2x cos x + 2 sin x; A1 x² sin x + 2x cos x − 2 sin x; M1 limits; A1 π²/4 − 2.

Q12Integration by substitution · Core · Paper 3 style · 5 marks

(a) Use the substitution u = 2x − 1 to find the exact value of ∫₁⁵ x/√(2x − 1) dx. [5]

Show answer

(a) 16/3
B1 x = (u+1)/2, dx = ½ du, limits u = 1 to 9; M1 integrand becomes ¼(u^{½} + u^{−½}); A1 ¼[(2/3)u^{3/2} + 2u^{½}]; M1 limits; A1 ¼(24 − 8/3) = 16/3.

Q13Integration by substitution · Standard · Paper 3 style · 6 marks

(a) Use the substitution u = 1 + x² to show that ∫₀^{√3} x³/√(1 + x²) dx = 4/3. [6]

Show answer

(a) 4/3
B1 du = 2x dx and x² = u − 1; B1 limits 1 and 4; M1 integrand ½(u − 1)u^{−½} = ½(u^{½} − u^{−½}); A1 ½[(2/3)u^{3/2} − 2u^{½}]; M1 limits; A1 ½(4/3 + 4/3) = 4/3.

Q14Integration by substitution · Challenge · Paper 3 style · 6 marks

(a) Use the substitution x = 2 sin θ to find the exact value of ∫₀¹ 1/(4 − x²)^{3/2} dx. [6]

Show answer

(a) √3/12 (≈ 0.144)
B1 dx = 2cos θ dθ; M1 (4 − 4sin²θ)^{3/2} = 8cos³θ; A1 integrand ¼ sec²θ; B1 limits 0 and π/6; M1 integrate to ¼ tan θ; A1 ¼ × 1/√3 = √3/12.

Q15Area between curves · Core · Paper 1 style · 6 marks

(a) Find the x-coordinates of the points where the line y = x + 3 meets the curve y = x² − 2x + 3. [2]
(b) Find the area of the region enclosed between the line and the curve. [4]

Show answer

(a) x = 0 and x = 3
M1 x² − 3x = 0; A1 x = 0, 3.
(b) 9/2
M1 ∫₀³ ((x + 3) − (x² − 2x + 3)) dx; A1 integrand 3x − x²; M1 [3x²/2 − x³/3]₀³; A1 27/2 − 9 = 9/2.

Q16Area between curves · Standard · Paper 1 style · 6 marks

(a) The curve y = 4√x and the line y = x meet at the origin O and at the point A. Find the coordinates of A. [2]
(b) Find the area of the region enclosed by the curve and the line. [4]

Show answer

(a) A(16, 16)
M1 4√x = x gives √x = 4; A1 (16, 16).
(b) 128/3 (≈ 42.7)
M1 ∫₀¹⁶ (4x^{½} − x) dx; A1 (8/3)x^{3/2} − x²/2; M1 limits; A1 512/3 − 128 = 128/3.

Q17Area between curves · Challenge · Paper 1 style · 8 marks

(a) The curve y = 4 − x² passes through P(1, 3). Find the equation of the normal to the curve at P. [3]
(b) The normal meets the curve again at Q. Find the x-coordinate of Q. [2]
(c) Find the area of the region enclosed by the curve and the normal. [3]

Show answer

(a) y = ½x + 5/2
B1 dy/dx = −2x = −2 at P; M1 normal gradient ½; A1 y − 3 = ½(x − 1).
(b) x = −3/2
M1 4 − x² = ½x + 5/2 gives 2x² + x − 3 = 0; A1 (2x + 3)(x − 1) = 0, x = −3/2.
(c) 125/48 (≈ 2.60)
M1 ∫_{−3/2}^{1} (3/2 − x/2 − x²) dx; A1 [3x/2 − x²/4 − x³/3]; A1 11/12 − (−27/16) = 125/48.

Q18Integration (ln forms) · Core · Paper 3 style · 3 marks

(a) Find the exact value of ∫₀¹ (4x + 2)/(x² + x + 1) dx, giving your answer as a single logarithm. [3]

Show answer

(a) 2 ln 3 = ln 9
M1 recognise 4x + 2 = 2 × d/dx(x² + x + 1); A1 2 ln(x² + x + 1); A1 2 ln 3 − 2 ln 1 = ln 9.

Q19Integration (ln forms) · Standard · Paper 2 style · 4 marks

(a) Show that (2x + 5)/(x + 1) can be written as 2 + 3/(x + 1). [1]
(b) Hence find the exact value of ∫₀² (2x + 5)/(x + 1) dx. [3]

Show answer

(a) Shown
B1 2(x + 1) + 3 = 2x + 5.
(b) 4 + 3 ln 3 (≈ 7.30)
M1 integrate to 2x + 3 ln(x + 1); A1; A1 (4 + 3 ln 3) − 0 = 4 + 3 ln 3.

Q20Integration of exponentials · Core · Paper 2 style · 5 marks

(a) Find ∫ (e^{2x} + 3e^{−x}) dx. [2]
(b) Hence show that ∫₀^{ln 2} (e^{2x} + 3e^{−x}) dx = 3. [3]

Show answer

(a) ½e^{2x} − 3e^{−x} + c
B1 ½e^{2x}; B1 −3e^{−x} (+ c).
(b) 3
M1 substitute ln 2: e^{2 ln 2} = 4, e^{−ln 2} = ½; A1 (2 − 3/2) − (½ − 3); A1 ½ + 5/2 = 3.

Q21Integration of exponentials · Challenge · Paper 2 style · 7 marks

(a) Given that ∫₀^k 4e^{2x} dx = 10, find the exact value of k. [3]
(b) Find the exact value of ∫₀¹ (eˣ + 1)² dx. [4]

Show answer

(a) k = ½ ln 6 (≈ 0.896)
M1 integrate to 2e^{2x}; M1 2e^{2k} − 2 = 10; A1 e^{2k} = 6, k = ½ ln 6.
(b) ½e² + 2e − 3/2 (≈ 7.63)
M1 expand e^{2x} + 2eˣ + 1; A1 ½e^{2x} + 2eˣ + x; M1 limits; A1 (½e² + 2e + 1) − (½ + 2) = ½e² + 2e − 3/2.

Q22Area / definite integrals · Core · Paper 1 style · 4 marks

(a) Evaluate ∫₁⁴ (3√x − 2/x²) dx. [4]

Show answer

(a) 25/2 = 12.5
M1 increase powers: 2x^{3/2} + 2x^{−1}; A1 each term correct (A1 A1 if separate); M1 limits (16 + ½) − (2 + 2); A1 12.5.

Q23Area / definite integrals · Challenge · Paper 1 style · 5 marks

(a) Find ∫ 4/(2x + 1)³ dx. [2]
(b) Hence find the value of the improper integral ∫₁^∞ 4/(2x + 1)³ dx, explaining how the upper limit is dealt with. [3]

Show answer

(a) −1/(2x + 1)² + c
M1 (2x + 1)^{−2} ÷ (−2 × 2) form; A1 −(2x + 1)^{−2} + c.
(b) 1/9
M1 as x → ∞, −1/(2x + 1)² → 0; M1 0 − (−1/9); A1 1/9.

Q24Volume of revolution · Standard · Paper 1 style · 4 marks

(a) The region bounded by the curve y = √(x + 1), the x-axis and the lines x = 0 and x = 3 is rotated through 360° about the x-axis. Find the exact volume of the solid formed. [4]

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(a) 15π/2
M1 V = π∫₀³ (x + 1) dx; A1 π[x²/2 + x]; M1 limits; A1 π(9/2 + 3) = 15π/2.

Q25Volume of revolution · Challenge · Paper 1 style · 7 marks

(a) The line y = 5 meets the curve y = x² + 1 at the points A and B. Find the x-coordinates of A and B. [1]
(b) The region enclosed between the curve and the line is rotated through 360° about the x-axis. Find the exact volume of the solid formed. [6]

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(a) x = −2 and x = 2
B1 x² = 4.
(b) 1088π/15 (≈ 228)
M1 V = π∫_{−2}^{2} (5² − (x² + 1)²) dx; M1 expand (x² + 1)² = x⁴ + 2x² + 1; A1 integrand 24 − 2x² − x⁴; A1 [24x − 2x³/3 − x⁵/5]; M1 limits (or 2 × from 0 to 2); A1 2π(48 − 16/3 − 32/5) = 1088π/15.

Q26Area under curve · Core · Paper 1 style · 6 marks

(a) The curve y = x³ − 6x² + 8x crosses the x-axis at O, A and B. Find the x-coordinates of A and B. [2]
(b) Find the total area of the regions enclosed between the curve and the x-axis from x = 0 to x = 4. [4]

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(a) x = 2 and x = 4
M1 x(x − 2)(x − 4) = 0; A1 2 and 4.
(b) 8
M1 integrate to x⁴/4 − 2x³ + 4x²; A1 ∫₀² = 4; A1 ∫₂⁴ = −4 (region below the axis); A1 total area 4 + 4 = 8.

Q27Area under curve · Standard · Paper 1 style · 6 marks

(a) Find the area of the region bounded by the curve y = 9/(x + 1)², the x-axis and the lines x = 0 and x = 2. [3]
(b) The line x = k divides this region into two parts of equal area. Find k. [3]

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(a) 6
M1 integrate to −9/(x + 1); M1 limits −3 − (−9); A1 6.
(b) k = ½
M1 9 − 9/(k + 1) = 3 (half of 6); A1 9/(k + 1) = 6; A1 k = ½.

Q28Integration using partial fractions · Standard · Paper 3 style · 5 marks

(a) Express 5/((x − 1)(x + 4)) in partial fractions. [2]
(b) Hence find the exact value of ∫₂⁴ 5/((x − 1)(x + 4)) dx, giving your answer as a single logarithm. [3]

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(a) 1/(x − 1) − 1/(x + 4)
M1 5 = A(x + 4) + B(x − 1); A1 A = 1, B = −1.
(b) ln(9/4)
M1 ln(x − 1) − ln(x + 4); M1 limits (ln 3 − ln 8) − (ln 1 − ln 6); A1 ln(18/8) = ln(9/4).

Q29Integration using partial fractions · Challenge · Paper 3 style · 9 marks

(a) Express (2x² + 3)/(x²(x + 1)) in partial fractions. [4]
(b) Hence show that ∫₁² (2x² + 3)/(x²(x + 1)) dx = 3/2 + 5 ln 3 − 8 ln 2. [5]

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(a) −3/x + 3/x² + 5/(x + 1)
B1 form A/x + B/x² + C/(x + 1); M1 2x² + 3 = Ax(x + 1) + B(x + 1) + Cx²; A1 B = 3, C = 5; A1 A = −3.
(b) 3/2 + 5 ln 3 − 8 ln 2 (≈ 1.448)
M1 integrate: −3 ln x − 3/x + 5 ln(x + 1); A1 (follow through); M1 limits: (−3 ln 2 − 3/2 + 5 ln 3) − (−3 + 5 ln 2); A1 3/2 + 5 ln 3 − 8 ln 2; A1 fully correct log work.

Q30Integration (ln and tan^-1 forms) · Challenge · Paper 3 style · 5 marks

(a) Find ∫ (2x + 3)/(x² + 1) dx. [3]
(b) Hence find the exact value of ∫₀¹ (2x + 3)/(x² + 1) dx. [2]

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(a) ln(x² + 1) + 3 tan⁻¹x + c
M1 split into 2x/(x² + 1) + 3/(x² + 1); A1 ln(x² + 1); A1 3 tan⁻¹x + c.
(b) ln 2 + 3π/4 (≈ 3.05)
M1 limits, tan⁻¹1 = π/4; A1 ln 2 + 3π/4.

Subtopics

Our analysis of how often each subtopic was examined and how many marks it carried, 2022 to May/June 2026.

Subtopic	Marks indexed	Sessions
Integration of trig functions Banker	170	9/9
Curve from gradient Banker	100	9/9
Integration by parts Banker	95	9/9
Integration by substitution Banker	86	9/9
Area between curves Banker	83	9/9
Integration (ln forms)	74	7/9
Integration of exponentials Banker	72	8/9
Area / definite integrals Banker	51	8/9
Volume of revolution	50	6/9
Area under curve	33	6/9
Integration using partial fractions	29	4/9
Integration (ln and tan^-1 forms) Rare	11	2/9

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