Methods in Calculus (Edexcel A Level Further Maths: Core Pure): Exam Questions

Show that

${\int }_0^{\infty }\frac{8x-12}{(2x^2+3)(x+1)}\mathrm{d}x=\ln k$

where $k$ is a rational number to be found.

$\mathrm{f}\left(x\right)=\frac{1}{\sqrt{4x^2+9}}$

Using a substitution, that should be stated clearly, show that

$\int \mathrm{f}\left(x\right)\mathrm{d}x=A{\sinh }^{-1}\left(Bx\right)+c$

where $c$ is an arbitrary constant and $A$ and $B$ are constants to be found.

Hence find, in exact form in terms of natural logarithms, the mean value of$\mathrm{f}\left(x\right)$ over the interval $[0,3]$.

$y={\tan }^{-1}x$

Assuming the derivative of $\tan x,$ prove that

$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{1+x^2}$

$\mathrm{f}\left(x\right)=x{\tan }^{-1}4x$

Show that

$\int \mathrm{f}\left(x\right)\mathrm{d}x=A{x}^2{\tan }^{-1}4x+Bx+C{\tan }^{-1}4x+k$

where $k$ is an arbitrary constant and $A,B$ and $C$ are constants to be determined.

Hence find, in exact form, the mean value of $\mathrm{f}\left(x\right)$ over the interval $\left[0,\frac{\sqrt{3}}{4}\right]$

$\mathrm{f}\left(x\right)=\frac{x+2}{x^2+9}$

Show that

$\int \mathrm{f}\left(x\right)\mathrm{d}x\mathit{=}A\ln (x^2+9)+B\arctan \left(\frac{x}{3}\right)+c$

where $c$ is an arbitrary constant and $A$ and $B$ are constants to be found.

Hence show that the mean value of $\mathrm{f}\left(x\right)$ over the interval $[0,3]$ is

$\frac{1}{6}\ln 2+\frac{1}{18}\pi$

Use the answer to part (b) to find the mean value, over the interval $[0,3]$, of

$\mathrm{f}\left(x\right)+\ln k$

where $k$ is a positive constant, giving your answer in the form $p+\frac{1}{6}\ln q,$ where$p$ and $q$ are constants and $q$ is in terms of $k$.

Explain why

${\int }_{-1}^{\infty }\frac{1}{x^2+6x+13} \text{d}x$

is an improper integral.

Show that

${\int }_{-1}^{\infty }\frac{1}{x^2+6x+13} \text{d}x=k\pi$

where $k$ is a constant to be determined.