Integration: past paper questions

Where the questions come from

• Question 1Oct/Nov 2025, paper 21Structured3 marks

  Find $\int 6{\sin }^{2}x\mathrm{d}x$.

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• Question 3Oct/Nov 2023, paper 21Structured6 marks

  Find ${\int }_4^{10}\frac{4}{2x-5}\mathrm{d}x$, giving your answer in the form $\ln a$, where $a$ is an integer.

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• Question 5Oct/Nov 2025, paper 21Structured7 marks

  The diagram shows the curve with equation $y=8{\mathrm{e}}^{-\frac{1}{2}x}-1$. The curve meets the axes at the points $A$ and $B$. The shaded region is bounded by the curve and the line segment $AB$.

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• Question 5Oct/Nov 2024, paper 21Structured8 marks

  It is given that ${\int }_a^{a^3}\frac{10}{2x+1}\mathrm{d}x=7$, where $a$ is a constant greater than 1. Show that $a=\sqrt[3]{0.5{\mathrm{e}}^{1.4}(2a+1)-0.5}$.

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• Question 2Oct/Nov 2024, paper 22Structured6 marks

  Let $\mathrm{f}(x)=4{\sin }^{2}3x$. Find the value of ${\mathrm{f}}^{\mathrm{\prime }}\left(\frac{1}{4}\pi \right)$.

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• Question 3Oct/Nov 2023, paper 22Structured5 marksHas a diagram

  The diagram shows the curve with equation $y=6{\mathrm{e}}^{-\frac{1}{2}x}$. The points on the curve with $x$-coordinates 0 and 2 are denoted by $A$ and $B$ respectively.

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• Question 7May/June 2026, paper 22Structured7 marksHas a diagram

  The diagram shows part of the curve with equation $y=8\cos 2x-\frac{1}{{\cos }^{2}2x}$. The shaded region is bounded by the curve and the $x$-axis. Find the exact area of the shaded region.

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• Question 5Oct/Nov 2023, paper 22Structured8 marks

  Find the quotient when $6x^3-5x^2-24x-4$ is divided by $(2x+1)$, and show that the remainder is 6.

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• Question 1May/June 2026, paper 23Structured4 marks

  Find ${\int }_a^{2a}(1+{\mathrm{e}}^{-x}{)}^2\mathrm{d}x$.

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• Question 4May/June 2025, paper 21Structured7 marksHas a diagram

  The diagram shows the curve with equation $y=6{\mathrm{e}}^{2x}-{\mathrm{e}}^{3x}$. The shaded region is bounded by the axes and the curve. Find the exact $x$-coordinate of the maximum point.

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• Question 3May/June 2023, paper 21Structured7 marks

  It is given that ${\int }_0^a(3{\mathrm{e}}^{2x}-1)\mathrm{d}x=12$, where $a$ is a positive constant. Show that $a=\frac{1}{2}\ln (9+\frac{2}{3}a)$.

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• Question 1May/June 2025, paper 22Structured3 marks

  Show that ${\int }_2^{11}\frac{8}{4x+1}\mathrm{d}x=\ln a$, where $a$ is an integer to be found.

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• Question 3May/June 2024, paper 22Structured8 marksHas a diagram

  The diagram shows the curve with equation $y=8{\mathrm{e}}^{-x}-{\mathrm{e}}^{2x}$. The curve crosses the $y$-axis at the point $A$ and the $x$-axis at the point $B$. The shaded region is bounded by the curve and the two axes.

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• Question 6May/June 2023, paper 21Structured7 marks

  Show that ${\int }_{\frac{1}{4}\pi }^{\frac{1}{3}\pi }(4{\cos }^{2}2x+\frac{1}{{\cos }^{2}x})\mathrm{d}x=\frac{3}{4}\sqrt{3}+\frac{1}{6}\pi -1$.

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• Question 4May/June 2025, paper 22Structured8 marksHas a diagram

  The diagram shows parts of the curves with equations $y=4{\mathrm{e}}^{-2x}$ and $y=1+0.5\sin 3x$. Point $P$ is a point of intersection of the curves, and the shaded region is bounded by the two curves and the $y$-axis.

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• Question 5May/June 2024, paper 22Structured8 marks

  The polynomial $\mathrm{p}(x)$ is defined by $\mathrm{p}(x)=9x^3+18x^2+5x+4$. Find the quotient when $\mathrm{p}(x)$ is divided by $(3x+2)$, and show that the remainder is 6.

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• Question 3May/June 2023, paper 22Structured5 marksHas a diagram

  The diagram shows part of the curve $y=\frac{6}{2x+3}$. 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 in the form $a-\ln b$, where $a$ and $b$ are integers.

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