Mathematics Extension 2

Find $A$ and $B$ such that
$\frac{15}{x^2-2x-15}=\frac A{x+3}+\frac B{x-5}.$

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

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

Find $\int\sqrt{\frac{x-1}{x+1}}\,dx.$

Using the substitution $x=\sec\theta$ or otherwise show by integrating that
$\int\frac{1}{\sqrt{x^2-1}}\,dx=\ln\left(x+\sqrt{x^2-1}\right).$

Hence find
$\int\frac{-7x-18}{(x-1)(x^2+4)}\,dx.$

Given the complex number $z=e^{i\theta}$, show that
$w=\frac{\bar z^2-1}{\bar z^2+1}$
is purely imaginary.

Using partial fractions, evaluate
$\int_2^n\frac{4+x}{(1-x)(4+x^2)}\,dx,$
giving your answer in the form
$\frac12\ln\left(\frac{f(n)}{8(n-1)^2}\right),$
where $f(n)$ is a function of $n$.

Hence, or otherwise, find $x$ as a function of $v$.

A particle with mass $1\,\mathrm{kg}$ is moving along the $x$-axis. Initially, the particle is at the origin and has speed $u\,\mathrm{m\,s^{-1}}$ to the right. The particle experiences a resistive force of magnitude $v+3v^2$ newtons, where $v\,\mathrm{m\,s^{-1}}$ is the speed of the particle after $t$ seconds. The particle is never at rest.

Show that
$\frac{dv}{dx}=-(1+3v).$

Consider the equation $z^5+1=0$, where $z$ is a complex number. Solve the equation $z^5+1=0$ by finding the fifth roots of $-1$.

Prove that for all integers $n$ with $n\ge3$, if $2^n-1$ is prime, then $n$ cannot be even.

The numbers $a_n$, for integers $n\ge1$, are defined as
$a_1=\sqrt2,\quad a_2=\sqrt{2+\sqrt2},\quad a_3=\sqrt{2+\sqrt{2+\sqrt2}},$
and so on.

These numbers satisfy the relation $a_{n+1}^2=2+a_n$, for $n\ge1$. (Do NOT prove this.)

Use mathematical induction to prove that
$a_n=2\cos\frac{\pi}{2^{n+1}},$ for all integers .

Show that if $z$ is a solution of $z^5+1=0$ and $z\ne-1$, then $u=z+\frac1z$ is a solution of $u^2-u-1=0$.

Hence, find the exact value of $\cos\frac{3\pi}{5}$.

Prove by contraposition that if $n^2$ is not divisible by $18$, then $n$ is not divisible by $6$.

Using (i), prove that $\sqrt{18}$ is irrational.

Prove that if $a$ is divisible by $5$, then $a^2-5a$ is divisible by $50$.

(You may assume that the product of two consecutive numbers is even.)

Prove by contradiction that $\sqrt2$ is irrational.

Prove for all real $a$ and $b$ that
$a^2+b^2\ge 2ab.$

Hence prove for all $a>0$ that
$a+\frac1a\ge2.$

Hence prove for all $a_1,a_2>0$ that
$(a_1+a_2)\left(\frac1{a_1}+\frac1{a_2}\right)\ge4.$

Hence prove by induction for all $a_1,a_2,a_3,\ldots,a_n>0$ that
$(a_1+a_2+\cdots+a_n)\left(\frac1{a_1}+\frac1{a_2}+\cdots+\frac1{a_n}\right)\ge n^2.$

(Note: this has already been proven for $n=1$ and $n=2$.)

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