Integration — AS & A Level Mathematics (9709) questions by topic
Evaluating a definite integral of a rational function to reach a logarithmic answer, finding the area under a curve involving fractional powers of x between two roots, and using integration to find an unknown constant from a given definite-integral value are typical of the 33 integration questions from 2023 to 2025, worth 3 to 12 marks.
Find, show, determine, use and deduce lead the command words, each demanding a fully shown antiderivative and substitution of limits rather than a calculator-only final value. Because a missing negative sign or a wrong limit substitution is the most common way marks are lost here, submit your integration working for instant marking, so any such slip is caught at the exact line it occurs.
May–June 2025, Paper 11
The diagram shows the curve with equation y = 5x^(3/2) − 20x and the line with equation y = x − 16. The x-coordinates of the points of intersection of the curve and line are 1 and…
Answer this question and get it marked →May–June 2025, Paper 12
The diagram shows the curve with equation y = 9 / (5x + 4)^(1/2) and the line y = 6 − 3x. The line and the curve intersect at the point P which has y-coordinate 3. The shaded…
Answer this question and get it marked →The equation of a curve is such that d^2y/dx^2 = −24/x^3. It is given that the curve has a stationary point at (−2, 19). Find an expression for dy/dx.
Answer this question and get it marked →May–June 2025, Paper 13
Given that the integral from 1 to 3 of ( a/((4x − 3)^2) + 2 ) dx = 12, find the value of the constant a.
Answer this question and get it marked →May–June 2025, Paper 15
The equation of a curve is such that dy/dx = 12(2x − 5)^2 + 8x. It is given that the curve passes through the point (2, 4). Find an equation of the curve. [4]
Answer this question and get it marked →The diagram shows part of the curve y = x^2 − 1/x^2. The shaded region is bounded by the curve, the line x = 2 and the x-axis. Find the volume formed when the shaded region is…
Answer this question and get it marked →May–June 2025, Paper 22
Show that integral from 2 to 11 of 8/(4x + 1) dx = ln a, where a is an integer to be found.
Answer this question and get it marked →May–June 2025, Paper 23
Show that integral from 2 to 11 of 8/(4x + 1) dx = ln a, where a is an integer to be found.
Answer this question and get it marked →May–June 2025, Paper 25
Show that integral from 2 to 11 of 8/(4x + 1) dx = ln a, where a is an integer to be found.
Answer this question and get it marked →May–June 2025, Paper 31
The constant a is such that ∫ from 1 to a of 6x ln x dx = 4. Show that a = exp((1/6)(5/a² + 3)), where exp(x) denotes e^x.
Answer this question and get it marked →May–June 2025, Paper 33
Find the exact value of the integral from (1/5)π to (1/4)π of 3 cos²(5x) dx, i.e. ∫{(1/5)π}^{(1/4)π} 3 cos²(5x) dx.
Answer this question and get it marked →May–June 2024, Paper 13
A curve passes through the point (4/5, −3) and is such that dy/dx = −20/(5x − 3)^2. Find the equation of the curve.
Answer this question and get it marked →May–June 2024, Paper 21
The diagram shows the curve with equation y = √(sin 2x + sin² 2x) for 0 ≤ x ≤ (1/6)π. The shaded region is bounded by the curve and the straight lines x = (1/6)π and y = 0. Use…
Answer this question and get it marked →May–June 2024, Paper 31
Use the substitution u = 1 − sin x to find the exact value of ∫ from π to (3/2)π of (sin 2x) / √(1 − sin x) dx. Give your answer in the form a + b√2, where a and b are rational…
Answer this question and get it marked →October–November 2024, Paper 21
It is given that ∫ₐ^(a³) 10/(2x + 1) dx = 7, where a is a constant greater than 1. Show that a = ∛(0.5 e^(1.4)(2a + 1) − 0.5). [5]
Answer this question and get it marked →October–November 2024, Paper 22
The diagram shows the curves with equations y = ∛(5x² + 7) and y = 27/(2x + 5) for x ≥ 0. The curves meet at the point (2, 3). Region A is bounded by the curve y = ∛(5x² + 7) and…
Answer this question and get it marked →October–November 2024, Paper 23
It is given that ∫ from a to a³ of 10/(2x + 1) dx = 7, where a is a constant greater than 1. Show that a = ³√(0.5 e^(1.4) (2a + 1) − 0.5).
Answer this question and get it marked →October–November 2024, Paper 31
Find the exact value of ∫₁³ x² ln 3x dx. Give your answer in the form a ln b + c, where a and c are rational and b is an integer.
Answer this question and get it marked →May–June 2023, Paper 11
The diagram shows part of the curve y = 4/(2x − 1)² and parts of the lines x = 1 and y = 1; the curve passes through A(1, 4) and B(3/2, 1), and the shaded region is bounded by the…
Answer this question and get it marked →May–June 2023, Paper 12
The equation of a curve is such that dy/dx = 4/(x − 3)³ for x 3. The curve passes through the point (4, 5). Find the equation of the curve.
Answer this question and get it marked →The diagram shows the curve with equation y = 10x^(1/2) − (5/2)x^(3/2) for x 0. The curve meets the x-axis at the points (0, 0) and (4, 0). Find the area of the shaded region.
Answer this question and get it marked →May–June 2023, Paper 13
A curve which passes through (0, 3) has equation y = f(x). It is given that f′(x) = 1 − 2/(x − 1)³. Find the equation of the curve.
Answer this question and get it marked →May–June 2023, Paper 21
It is given that ∫₀ᵃ (3e^(2x) − 1) dx = 12, where a is a positive constant. Show that a = (1/2) ln(9 + (2/3)a). [4]
Answer this question and get it marked →Show that ∫ from (1/4)π to (1/3)π of (4 cos^2 2x + 1/cos^2 x) dx = (3/4)√3 + (1/6)π − 1. [7]
Answer this question and get it marked →May–June 2023, Paper 22
The diagram shows part of the curve y = 6/(2x + 3). The shaded region is bounded by the curve and the lines x = 6 and y = 2. Find the exact area of the shaded region, giving your…
Answer this question and get it marked →May–June 2023, Paper 23
The diagram shows part of the curve y = 6 / (2x + 3). The shaded region is bounded by the curve and the lines x = 6 and y = 2. Find the exact area of the shaded region, giving…
Answer this question and get it marked →May–June 2023, Paper 31
The constant a is such that ∫₀ᵃ x e^(−2x) dx = 1/8. Show that a = ½ ln(4a + 2).
Answer this question and get it marked →May–June 2023, Paper 33
Use the substitution u = cos x to show that ∫₀^π sin 2x e^(2 cos x) dx = ∫₋₁¹ 2u e^(2u) du.
Answer this question and get it marked →October–November 2023, Paper 11
A curve has a stationary point at (2, −10) and is such that d²y/dx² = 6x. Find dy/dx.
Answer this question and get it marked →October–November 2023, Paper 13
A curve is such that its gradient at a point (x, y) is given by dy/dx = x − 3x^(−1/2). It is given that the curve passes through the point (4, 1). Find the equation of the curve.
Answer this question and get it marked →October–November 2023, Paper 21
Find ∫₄¹⁰ 4/(2x − 5) dx, giving your answer in the form ln a, where a is an integer. [4]
Answer this question and get it marked →October–November 2023, Paper 23
Find ∫₄¹⁰ 4/(2x − 5) dx, giving your answer in the form ln a, where a is an integer.
Answer this question and get it marked →October–November 2023, Paper 32
Find the exact value of ∫₀⁶ [x(x + 1) / (x² + 4)] dx.
Answer this question and get it marked →Other AS & A Level Mathematics topics
- Differentiation (95)
- Algebra, logarithmic and exponential functions (87)
- Trigonometry (57)
- Series (37)
- Kinematics (33)
- The normal distribution (32)
- Energy, work and power (31)
- The Poisson distribution (31)
- Complex numbers (28)
- Sampling and estimation (28)
- Permutations and combinations (27)
- Representation of data (23)