Integration (AQA A Level Maths: Pure): Exam Questions

Find the value of the positive constant $k$ such that

${\int }_k^5(2x-1)\mathrm{d}x=20$

Find

$\int \left(x^2+5-\frac{3}{x^2}\right)\mathrm{d}x$

giving your answer in simplest form.

Write down

$\int 3{\mathrm{e}}^{3x}dx$

The figure below shows a sketch of the line $y=2$ and the curve with equation $y=x^2-4x+5$.

Use algebraic integration to find the exact area of $R$.

The curve with equation $y=x^2$ and the straight line with equation $y=6-x$ are shown in the figure below.

The finite region bounded by the curve, the line and the $x$-axis is shaded.

Use algebraic integration to find the exact area of the shaded region.

Use algebraic integration to find the exact value of

${\int }_4^9\frac{x^2+1}{\sqrt{x}}\mathrm{d}x$

The finite region $R$, shown in the figure below, is bounded by the curve with equation$y=4-{\left(x-2\right)}^2$ and the $x$-axis.

The $x$-intercepts of the curve $y=4-{\left(x-2\right)}^2$ are 0 and 4.

Use algebraic integration to find the exact area of $R$.

Find

$\int \left(2\sqrt{x}+5x^{\frac{1}{3}}\right)\mathrm{d}x$

giving your answer in simplest form.

A curve has the equation $y=\mathrm{f}\left(x\right)$

Given that

• ${\mathrm{f}}^{\text{'}}\left(x\right)=3-2x+\frac{4}{x^2}$

• the curve passes through the point $\left(-2,3\right)$

find the equation of the curve, giving your answer in simplest form.

Find the integer value of $b$ such that

${\int }_{-3}^4\left(2kx+3k{x}^2\right)\mathrm{d}x=bk$

where $k$ is a constant.

A curve has the equation $y=\mathrm{f}\left(x\right)$

Given that

• $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=30x$

• $\frac{dy}{dx}=17$ when $x=1$

find an expression for $\frac{dy}{dx}$

Given that

• $y=40$ when $x=2$

find the equation of the curve, $y=\mathrm{f}\left(x\right)$, giving your answer in simplest form.

The figure below shows a sketch of the curve with equation $y=x(x-1)(x+2)$.

Use algebraic integration to find the total area of the shaded regions.

Use algebraic integration to find the value of

${\int }_2^4\frac{x^3+\sqrt[3]{x}}{2\sqrt{x}}dx$

giving your answer correct to 3 significant figures.

Find

$\int {\left(2-x\right)}^3\mathrm{d}x$

writing each term in simplest form.

The function $\mathrm{f}\left(x\right)$ has the following properties

• ${\mathrm{f}}^{\text{'}\text{'}}\left(x\right)=6(x-2)$

• ${\mathrm{f}}^{\text{'}}\left(2\right)=8$

• $\mathrm{f}\left(3\right)=20$

Find $\mathrm{f}\left(x\right)$, giving your answer in simplest form.

The figure below shows the straight line with equation $5y=14-3x$ and the curve with equation $5y=20-2x-x^2$.

The shaded region $R$ is bounded by the curve and the straight line.

Use algebraic integration to find the exact area of $R$.

The curve with equation $y=x^3-2x^2-x+2$ is shown in the figure below.

Use algebraic integration to find the total area of the shaded regions.

Given that

$\int {\left(2\left(3-\frac{1}{2}x\right)\right)}^p\mathrm{d}x=-\frac{1}{4}{x}^4+...$

where $p$ is a positive integer and $-\frac{1}{4}{x}^4$ is the term with the highest power of $x$, find fully

$\int {\left(2\left(3-\frac{1}{2}x\right)\right)}^p\mathrm{d}x$

writing each term in simplest form.

Find the value of the constant $q$ such that

${\int }_q^{4q}5x\sqrt{x}\mathrm{d}x=15066$

A function, $\mathrm{f}\left(x\right)$

• has a factor of $\left(2x-1\right)$

• has a factor of $\left(3x+2\right)$

• has a second derivative of $\mathrm{f}\text{'}\text{'}\left(x\right)=2\left(18x-5\right)$

Find $\mathrm{f}\left(x\right)$.

The straight line with equation $4y=4x+17$ and the curve with equation $y=3x^2-8x-16$ are shown in the figure below.

The region bounded by the straight line, the curve and the $x$-axis is shaded.

Use algebraic integration to find the exact area of the shaded region.