Mathematics Extension 2

Hence, or otherwise, find
$\int\frac{x^2-2x}{x^2-2x-15}\,dx.$

Evaluate
$\int_0^{\pi/4}\sec^4x\tan^2x\,dx.$

Use the substitution $x=2\sin\theta$ to find
$\int\frac{dx}{(4-x^2)^{3/2}}.$

Use the substitution $t=\tan\frac x2$, or otherwise, to evaluate
$\int_0^{\pi/2}\frac{1}{3\sin x-4\cos x+5}\,dx.$

Find $\displaystyle\int\cos^{-1}x\,dx$.

Let $J_n=\int_0^1x^ne^{-x}\,dx$. Using parts (ii) and (iv), prove that
$e=\lim_{n\to\infty}\sum_{r=0}^{n}\frac1{r!}.$

Let $J_n=\int_0^1x^ne^{-x}\,dx$, where $n$ is a non-negative integer. Using parts (i) and (iii), show by mathematical induction, or otherwise, that for all $n\ge0$,
$J_n=n!-\frac{n!}{e}\sum_{r=0}^{n}\frac1{r!}.$

Let
$J_n=\int_0^1x^ne^{-x}\,dx,$
where $n$ is a non-negative integer. Show that
$J_0=1-\frac1e.$

Use a suitable substitution to show that
$\int_0^a f(x)\,dx=\int_0^a f(a-x)\,dx.$

Let $J_n=\int_0^1x^ne^{-x}\,dx$, where $n$ is a non-negative integer. Show that
$J_n=nJ_{n-1}-\frac1e,\qquad n\ge1.$

Let $J_n=\int_0^1x^ne^{-x}\,dx$, where $n$ is a non-negative integer. Show that
$J_n\le\frac1{n+1}.$

Hence, or otherwise, evaluate
$\int_0^{\pi/4}\frac{\sin x}{\cos x+\sin x}\,dx.$

Consider
$I_n=\int_0^1\frac{x^n}{\sqrt{1-x}}\,dx.$
Use integration by parts to show that
$I_n=\frac{2n}{2n+1}I_{n-1}.$

Using the substitution $x=\tan^2\theta$, evaluate
$\int_0^1\sin^{-1}\sqrt{\frac{x}{1+x}}\,dx.$

Hence find
$\int_0^{\pi/2}\cos^7\theta\,d\theta.$

Find
$\int\frac{\ln x-1}{(x+\ln x)^2}\,dx.$

If
$I_n=\int_0^{\pi/2}\cos^n\theta\,d\theta,$
show that
$I_n=\frac{n-1}{n}I_{n-2}.$

If
$I_n=\int_0^{\pi/2}\cos^n\theta\,d\theta,$
hence find
$\int_0^{\pi/2}\cos^7\theta\,d\theta.$

Which expression is equal to $\int x^5e^{7x}\,dx$?

A. $\frac17x^5e^{7x}-\frac57\int x^4e^{7x}\,dx$
B. $\frac17x^5e^{7x}-\frac57\int x^5e^{7x}\,dx$
C. $\frac57x^4e^{7x}-\frac57\int x^4e^{7x}\,dx$
D. $\frac57x^4e^{7x}-\frac57\int x^5e^{7x}\,dx$

Express $\frac{3x^2-5}{(x-2)(x^2+x+1)}$ as a sum of partial fractions over $\mathbb R$.

Find $\int\frac{2x+3}{x^2+2x+2}\,dx$.

Region $A$ is bounded by $y=1$ and $x^2+y^2=1$ between $x=0$ and $x=1$. Region $B$ is bounded by $y=1$ and $y=\ln x$ between $x=1$ and $x=e$. Using part (i), or otherwise, show that the ratio of the volumes formed by rotating regions $A$ and $B$ about the $x$-axis is $V_A:V_B=1:3$.

An object of mass $1\,\mathrm{kg}$ is projected vertically upwards with initial velocity $u\,\mathrm{m\,s^{-1}}$. It experiences air resistance of magnitude $kv^2$ newtons, where $v$ is velocity in $\mathrm{m\,s^{-1}}$ and $k>0$. The height above its starting point is $x$ metres, time is $t$ seconds, and acceleration due to gravity is $g\,\mathrm{m\,s^{-2}}$. Show that the time to reach maximum height is
$\frac1{\sqrt{gk}}\arctan\left(u\sqrt{\frac{k}{g}}\right)\text{ seconds}.$

For the resisted vertical motion described in part (i), find an expression for the maximum height reached by the object, in terms of $k$, $g$ and $u$.