Integration: past paper questions

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• Question 2May/June 2026, paper 21Structured7 marks

  Let $I_n={\int }_0^1(1+t{)}^n\sin \pi t\mathrm{d}t$, where $n$ is a non-negative integer. Show that, for $n⩾2$, ${\pi }^2{I}_n=\pi (2^n+1)-n(n-1)I_{n-2}$.

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• Question 3Oct/Nov 2025, paper 21Structured9 marks

  The integral $I_n$ is defined by $I_n={\int }_0^1(1+x^2{)}^n\mathrm{d}x$. By considering $\frac{\mathrm{d}}{\mathrm{d}x}(x(1+x^2{)}^n)$, or otherwise, show that $$(2n+1)I_n=2^n+2n{I}_{n-1}\mathrm{.}$$ [5]

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• Question 8Oct/Nov 2024, paper 21Structured14 marks

  By considering the binomial expansion of (z + 1/z )⁷, where z = cosθ + i sinθ, use de Moivre's theorem to show that cos⁷θ = a cos 7θ + b cos 5θ + c cos 3θ + d…

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• Question 8Oct/Nov 2023, paper 21Structured15 marksHas a diagram

  State the sum of the series $1+z+z^2+\dots +z^{n-1}$, for $z\mathrm{\ne }1$.

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• Question 2May/June 2026, paper 22Structured7 marks

  Let $I_n={\int }_0^1(1+t{)}^n\sin \pi t\mathrm{d}t$, where $n$ is a non-negative integer. Show that, for $n⩾2$, ${\pi }^2{I}_n=\pi (2^n+1)-n(n-1)I_{n-2}$.

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• Question 4Oct/Nov 2025, paper 22Structured8 marks

  By considering the binomial expansion of (z + 1/z )⁶, where z = cosθ + i sinθ, use de Moivre's theorem to show that cos⁶θ = a cos 6θ + b cos 4θ + c cos 2θ + d…

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• Question 1Oct/Nov 2024, paper 22Structured4 marks

  Find the value of ${\int }_6^7\frac{1}{\sqrt{(x-5{)}^2-1}}\mathrm{d}x$, giving your answer in the form $\ln (a+\sqrt{b})$, where $a$ and $b$ are integers to be determined.

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• Question 7Oct/Nov 2025, paper 22Structured11 marks

  Show that $\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{2}x\sqrt{4-x^2}+2{\sin }^{-1}\left(\frac{1}{2}x\right))=\sqrt{4-x^2}$.

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• Question 3May/June 2026, paper 23Structured6 marks

  Let $I_n={\int }_0^a(\ln (x+1){)}^n\mathrm{d}x$, where $n$ is a non-negative integer and $a>0$. By considering the derivative of $(x+1)(\ln (x+1){)}^n$, or otherwise, show that for $n⩾1$ $$I_n=(a+1)(\ln (a+1){)}^n-n{I}_{n-1}\mathrm{.}$$

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• Question 3Oct/Nov 2025, paper 23Structured9 marks

  The integral $I_n$ is defined by $I_n={\int }_0^1(1+x^2{)}^n\mathrm{d}x$. By considering $\frac{\mathrm{d}}{\mathrm{d}x}(x(1+x^2{)}^n)$, or otherwise, show that $$(2n+1)I_n=2^n+2n{I}_{n-1}\mathrm{.}$$

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• Question 7May/June 2026, paper 23Structured12 marksHas a diagram

  Use the substitution $u=1-x^2$ to find $\int \frac{x}{\sqrt{1-x^2}}\mathrm{d}x$.

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• Question 7Oct/Nov 2024, paper 23Structured10 marks

  Show that an appropriate integrating factor for $$\sqrt{x^2+16}\frac{\mathrm{d}y}{\mathrm{d}x}+y=x\sqrt{x^2+16}$$ is $\frac{1}{4}x+\frac{1}{4}\sqrt{x^2+16}$.

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• Question 3May/June 2023, paper 21Structured8 marks

  By considering the binomial expansion of $(z+z^{-1}{)}^4$, where $z=\cos \theta +\mathrm{i}\sin \theta$, use de Moivre's theorem to show that ${\cos }^4\theta =\frac{1}{8}(\cos 4\theta +4\cos 2\theta +3)$.

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• Question 4May/June 2023, paper 21Structured9 marks

  The integral $I_n$ is defined by $I_n={\int }_0^1(1+x^5{)}^n\mathrm{d}x$. By considering $\frac{\mathrm{d}}{\mathrm{d}x}(x(1+x^5{)}^n)$, or otherwise, show that $$(5n+1)I_n=2^n+5n{I}_{n-1}\mathrm{.}$$

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• Question 7May/June 2023, paper 21Structured11 marksHas a diagram

  Use the substitution $u=x^2-1$ to find $\int \frac{x}{\sqrt{x^2-1}}\mathrm{d}x$.

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• Question 7May/June 2025, paper 22Structured10 marks

  Find the solution of the differential equation $$\frac{\mathrm{d}y}{\mathrm{d}x}-\frac{x+5}{x^2+10x+61}y=1,$$ given that $y=0$ when $x=3$.

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• Question 7May/June 2024, paper 21Structured12 marks

  Use the substitution $u=1+x^2$ to find $$\int \frac{x}{\sqrt{1+x^2}}\mathrm{d}x\mathrm{.}$$

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• Question 7May/June 2024, paper 22Structured12 marks

  Use the substitution $u=1+x^2$ to find $$\int \frac{x}{\sqrt{1+x^2}}\mathrm{d}x\mathrm{.}$$

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• Question 7May/June 2023, paper 23Structured11 marks

  The integral $I_n$, where $n$ is an integer, is defined by $I_n={\int }_0^{\frac{4}{3}}(1+x^2{)}^{\frac{1}{2}n}\mathrm{d}x$. Find the exact value of $I_{-1}$ giving your answer in the form $\ln a$, where $a$ is an integer to be determined.

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• Question 1May/June 2024, paper 23Structured5 marks

  Find the exact value of ${\int }_2^{\frac{7}{2}}\frac{1}{\sqrt{4x-x^2-1}}\mathrm{d}x$.

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