Integration (Cambridge International AS Mathematics): Exam Questions

Integrate

(i) $2 x$

(ii) $6 x^{2}$

(iii) $\frac{1}{2} x^{\frac{1}{2}}$

Evaluate

(i) \int_{1}^{2} 4x \,\text{d}x

(ii) \int_{0}^{3} \left(9x^{2} + 4x\right) \,\text{d}x

Use calculus to find

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

Show that

$\frac{3x^{3}+4x^{6}}{x^{2}}$

can be written as $3 x^{a} + 4 x^{b}$, where $a$ and $b$ are constants to be found.

Hence find

\int \left(\dfrac{3x^{3} + 4x^{6}}{x^{2}}\right) \,\text{d}x .

Integrate $5 x^{4} + 6 x^{2} + 2 x + 3 .$

Given that $\text{f} ' ( x ) = 5 x^{4} + 6 x^{2} + 2 x + 3$ and that the graph of $y = \text{f} ( x )$ passes through the point $( 1 , 10 )$, find an expression for $\text{f} ( x )$ in terms of $x$ only.

Evaluate

\int_{-3}^{4} \left(2kx + 3kx^{2}\right) \,\text{d}x ,

giving your answer in terms of $k$.

A curve is such that $\frac{\text{d}y}{\text{d}x} = 2 x^{3} - x$. Given that the curve passes through the point $( 2 , - 2 )$, show that $2 y = x^{4} - x^{2} - 16$.

By first writing $\frac{1}{x^{2}}$ as $x^{-2}$, use calculus to show that

\int_{1}^{a} \dfrac{1}{x^{2}} \,\text{d}x = 1 - \dfrac{1}{a} ,

where $a > 1$ is a constant.

Find the value of the integral in part (a) when

(i) $a = 10$

(ii) $a = 1000$

(iii) $a = 1 000 000$

By considering your answers to part (b), suggest what number the value of the integral in part (a) will approach as $a \rightarrow \infty$.

Use calculus to find

\int \left(3x^{2} + 5x + 3\right) \,\text{d}x .

Evaluate

\int_{1}^{5} \left(4x + 6x^{2}\right) \,\text{d}x .

Show that

\left(3 - 2x\right)^{2} = 9 - 12x + 4x^{2} .

Hence, or otherwise, work out

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

Using the binomial expansion, or otherwise, show that

\left(2 - x\right)^{3} = 8 - 12x + 6x^{2} - x^{3} .

Hence, or otherwise, work out

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

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 $\frac{\text{d}y}{\text{d}x} = 3 - 2 x + 6 x^{2}$. 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 $\frac{\text{d}^{2}y}{\text{d}x^{2}} = 30 x$ and that when $x = 1$, $\frac{\text{d}y}{\text{d}x} = 17$, show that

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

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

Show that

\int_{1}^{a} \dfrac{2}{x^{3}} \,\text{d}x = 1 - \dfrac{1}{a^{2}} ,

where $a > 1$ is a constant.

Hence find the value of

\int_{1}^{\infty} \dfrac{2}{x^{3}} \,\text{d}x .

Find the value of

\int_{4}^{9} \dfrac{x^{2} + 1}{\sqrt{x}} \,\text{d}x .

A curve is such that $\frac{\text{d}y}{\text{d}x} = 3 - 2 x + \frac{4}{x^{2}}$. 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 + \dfrac{1}{x^{2}}\right) \,\text{d}x = \dfrac{15}{4} ,

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

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

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

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

Use calculus to find the value of

\int_{2}^{4} \dfrac{x^{3} + \sqrt[3]{x}}{2\sqrt{x}} \,\text{d}x ,

giving your answer correct to 3 significant figures.

Show that

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

Hence, or otherwise, work out

\int \left(2\left(3 - \dfrac{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} \dfrac{3}{x\sqrt{x}} \,\text{d}x = \sqrt{3} ,

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