Integration: past paper questions

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• Question 8May/June 2026, paper 31Structured6 marks

  Find the exact value of ${\int }_0^{1}3x^2{\mathrm{e}}^{3x}\mathrm{d}x$.

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• Question 1Oct/Nov 2025, paper 31Structured4 marks

  Find the exact value of ${\int }_1^2\ln 3x\mathrm{d}x$.

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• Question 2Oct/Nov 2024, paper 31Structured5 marks

  Find the exact value of ${\int }_1^3{x}^2\ln 3x\mathrm{d}x$.

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• Question 7Oct/Nov 2023, paper 31Structured7 marks

  The variables $x$ and $\theta$ satisfy the differential equation $$\frac{x}{\tan \theta }\frac{\mathrm{d}x}{\mathrm{d}\theta }=x^2+3.$$ It is given that $x=1$ when $\theta =0$. Solve the differential equation, obtaining an expression for $x^2$ in terms of $\theta$.

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• Question 8Oct/Nov 2025, paper 31Structured8 marks

  The variables $x$ and $y$ satisfy the differential equation $$(x^2+1)\frac{\mathrm{d}y}{\mathrm{d}x}=kx{\mathrm{e}}^{2y},$$ where $k$ is a constant. It is given that $y=0$ when $x=0$ and that $y=-\frac{1}{2}$ when $x=1$.

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• Question 6May/June 2026, paper 33Structured6 marks

  Use the substitution $u=\sqrt{x}$ to find the exact value of ${\int }_{\frac{1}{16}{\pi }^2}^{\frac{1}{4}{\pi }^2}\frac{\cos (3\sqrt{x})}{\sqrt{x}}\mathrm{d}x$.

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• Question 4Oct/Nov 2025, paper 32Structured6 marks

  Find the exact value of ${\int }_0^{1}x{\tan }^{-1}x\mathrm{d}x$.

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• Question 5Oct/Nov 2023, paper 32Structured6 marks

  Find the exact value of ${\int }_0^6\frac{x(x+1)}{x^2+4}\mathrm{d}x$.

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• Question 10May/June 2026, paper 33Structured8 marks

  The variables $x$ and $y$ satisfy the differential equation $$4{\mathrm{e}}^{2x}\frac{\mathrm{d}y}{\mathrm{d}x}=x(y+3)$$ for $y>-3$. It is given that $y=-2$ when $x=0$. Solve the differential equation and obtain an expression for $y$ in terms of $x$.

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• Question 9Oct/Nov 2024, paper 33Structured8 marks

  Find the quotient and remainder when $x^4+16$ is divided by $x^2+4$.

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• Question 9Oct/Nov 2025, paper 32Structured9 marks

  Let $\mathrm{f}(x)=\frac{x^2+4ax+6a^2}{(x+2a)(x+3a)}$, where $a$ is a positive constant. Express $\mathrm{f}(x)$ in partial fractions.

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• Question 8Oct/Nov 2023, paper 33Structured7 marks

  The variables $x$ and $y$ satisfy the differential equation $${\mathrm{e}}^{4x}\frac{\mathrm{d}y}{\mathrm{d}x}={\cos }^{2}3y\mathrm{.}$$ It is given that $y=0$ when $x=2$. Solve the differential equation, obtaining an expression for $y$ in terms of $x$.

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• Question 6Oct/Nov 2025, paper 33Structured6 marks

  Find the exact value of ${\int }_0^{\frac{1}{6}\pi }{x}^2\sin 2x\mathrm{d}x$.

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• Question 8May/June 2024, paper 31Structured7 marks

  Use the substitution $u=1-\sin x$ to find the exact value of $${\int }_{\pi }^{\frac{3}{2}\pi }\frac{\sin 2x}{\sqrt{1-\sin x}}\mathrm{d}x\mathrm{.}$$ Give your answer in the form $a+b\sqrt{2}$ where $a$ and $b$ are rational numbers to be determined.

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• Question 10May/June 2024, paper 31Structured10 marks

  Given that $2x=\tan y$, show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{2}{1+4x^2}$.

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• Question 9May/June 2025, paper 31Structured10 marks

  The constant $a$ is such that ${\int }_1^{a}6x\ln x\mathrm{d}x=4$. Show that $a=\exp \left(\frac{1}{6}(\frac{5}{a^2}+3)\right)$, where $\exp (x)$ denotes ${\mathrm{e}}^x$.

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• Question 6May/June 2024, paper 32Structured9 marksHas a diagram

  The diagram shows the curve $y=x{\mathrm{e}}^{-ax}$, where $a$ is a positive constant, and its maximum point $M$. Find the exact coordinates of $M$.

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• Question 8May/June 2023, paper 31Structured10 marks

  Let $\mathrm{f}(x)=\frac{3-3x^2}{(2x+1)(x+2{)}^2}$. Express $\mathrm{f}(x)$ in partial fractions.

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• Question 9May/June 2023, paper 31Structured10 marks

  The constant $a$ is such that ${\int }_0^{a}x{\mathrm{e}}^{-2x}\mathrm{d}x=\frac{1}{8}$. Show that $a=\frac{1}{2}\ln (4a+2)$.

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• Question 10May/June 2025, paper 32Structured8 marksHas a diagram

  Find the quotient and remainder when $x^2$ is divided by $1+4x^2$.

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• Question 9May/June 2023, paper 32Structured10 marks

  Let $\mathrm{f}(x)=\frac{2x^2+17x-17}{(1+2x)(2-x{)}^2}$. Express $\mathrm{f}(x)$ in partial fractions.

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• Question 8May/June 2024, paper 33Structured8 marksHas a diagram

  Express $3\cos 2x-\sqrt{3}\sin 2x$ in the form $R\cos (2x+\alpha )$, where $R>0$ and $0<\alpha <\frac{1}{2}\pi$.

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• Question 10May/June 2023, paper 32Structured10 marksHas a diagram

  The diagram shows the curve $y=(x+5)\sqrt{3-2x}$ and its maximum point $M$. Find the exact coordinates of $M$.

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• Question 10May/June 2025, paper 33Structured10 marksHas a diagram

  The variables $x$ and $y$ satisfy the differential equation $$\sin 4y\frac{\mathrm{d}y}{\mathrm{d}x}=x\sin 2y\sin 3x\mathrm{.}$$ It is given that $y=\frac{1}{12}\pi$ when $x=\frac{1}{2}\pi$. Solve the differential equation, obtaining a relation between $x$ and $y$.

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• Question 7May/June 2023, paper 33Structured8 marks

  Use the substitution $u=\cos x$ to show that $${\int }_0^{\pi }\sin 2x{\mathrm{e}}^{2\cos x}\mathrm{d}x={\int }_{-1}^{1}2u{\mathrm{e}}^{2u}\mathrm{d}u\mathrm{.}$$

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• Question 8May/June 2023, paper 33Structured8 marks

  The variables $x$ and $y$ satisfy the differential equation $$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{y^2+4}{x(y+4)}$$ for $x>0$. It is given that $x=4$ when $y=2\sqrt{3}$. Solve the differential equation to obtain the value of $x$ when $y=2$.

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