Integration (Cambridge (CIE) AS Maths: Pure 1): Exam Questions

A curve is such that $\frac{\text{d}y}{\text{d}x}=\frac{2}{\sqrt{x}}-x-3$. Given that the curve passes through the point $(4,-8)$, find the equation of the curve.

A curve is such that $\frac{\text{d}y}{\text{d}x}=3-2x+6x^2$. Given that the curve passes through the point $(4,64)$, find the equation of the curve.

Given that

${\int }_k^5\left(2x-1\right) \text{d}x=20,$

where $k$ is a positive constant, find the value of $k$.

Given $\frac{{\text{d}}^{2}y}{\text{d}{x}^2}=30x$ and that when $x=1$, $\frac{\text{d}y}{\text{d}x}=17$, show that

$\frac{\text{d}y}{\text{d}x}=15x^2+2.$

Find an equation for $y$ in terms of $x$, given that when $x=2$, $y=40$.

Show that

${\int }_1^a\frac{2}{x^3} \text{d}x=1-\frac{1}{a^2},$

where $a>1$ is a constant.

Hence find the value of

${\int }_1^{\infty }\frac{2}{x^3} \text{d}x.$

Find the value of

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

A curve is such that $\frac{\text{d}y}{\text{d}x}=3-2x+\frac{4}{x^2}$. Given that the curve passes through the point $(-2,3)$, find the equation of the curve.

Use calculus to find the exact value of the following improper integral:

${\int }_7^{\infty }\frac{5}{x^2}\text{d}x$

Use calculus to find

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

Given that

${\int }_1^p\left(1+\frac{1}{x^2}\right) \text{d}x=\frac{15}{4},$

where $p$ is a positive constant, find the value of $p$.

A function $\text{f}\left(x\right)$ has second derivative given by

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

Given that $\text{f}\left(3\right)=20$ and $\text{f}\text{'}\left(2\right)=8$, find $\text{f}\left(x\right)$.

Use calculus to find the value of

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

giving your answer correct to 3 significant figures.

Show that

${\left(3-\frac{1}{2}x\right)}^3=27-\frac{27}{2}x+\frac{9}{4}{x}^2-\frac{1}{8}{x}^3.$

Hence, or otherwise, work out

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

Given that

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

find the value of the constant $q$.

Given that

${\int }_p^{\infty }\frac{3}{x\sqrt{x}} \text{d}x=\sqrt{3},$

where $p$ is a real constant, find the value of $p$.