Mathematics Extension 2

Which expression is equal to

(A) $\displaystyle 2x\sqrt{x}\left(\ln x-\frac{2}{3}\right)+c$

(B) $\displaystyle 2x\sqrt{x}\left(\ln x+\frac{2}{3}\right)+c$

(C) $\displaystyle \frac{1}{\sqrt{x}}\left(\frac{3}{2}\ln x-1\right)+c$

(D) $\displaystyle \frac{1}{\sqrt{x}}\left(\frac{3}{2}\ln x+1\right)+c$

Show that

Let

and

for $n=0,1,2,3,\ldots$.

Hence find the value of $J_3$ in simplest exact form.

Let

and

for $n=0,1,2,3,\ldots$.

Show that $I_n=1-nI_{n-1}$ for $n=1,2,3,\dots$.

By making the substitution $u=a-x$, show that

Hence or otherwise, evaluate

Let

and

for $n=0,1,2,3,\ldots$.

Show that

for $n=0,1,2,3,\ldots$.

Find real numbers $a$, $b$ and $c$ such that

Hence find

Of the following expressions, which one need NOT contain a term involving a logarithm in its anti-derivative?

A. $\dfrac{x+2}{x^2+4x+5}$

B. $\dfrac{x+2}{x^2-4x-5}$

C. $\dfrac{x-1}{x^3-x^2+x-1}$

D. $\dfrac{x+1}{x^3-x^2+x-1}$

If $\displaystyle\int_a^x f(t)\,dt=g(x)$, which of the following is a primitive of $f(x)g(x)$?

A. $\frac12[f(x)]^2$

B. $\frac12[f'(x)]^2$

C. $\frac12[g(x)]^2$

D. $\frac12[g'(x)]^2$

$\int\frac{1}{\sqrt{6x-x^2}}\,dx=$

A. $\sin^{-1}\left(\frac{x}{\sqrt6}\right)+C$

B. $\sin^{-1}\left(\frac{x}{6}\right)+C$

C. $\sin^{-1}\left(\frac{x-3}{\sqrt6}\right)+C$

D. $\sin^{-1}\left(\frac{x-3}{3}\right)+C$

If $f(x)$ is even then
$\int_{-a}^{a}f(x)\,dx=$

A. $2\int_0^a f(a-x)\,dx$

B. $2\int_0^{-a} f(a-x)\,dx$

C. $2\int_a^0 f(x)\,dx$

D. All of the above.

Which of the following definite integrals has a positive value?

A. $\displaystyle\int_{-1/2}^{1/2}x^3(1+\cos x)\,dx$

B. $\displaystyle\int_{-1/2}^{1/2}\frac{\sin^{-1}x}{1+x^4}\,dx$

C. $\displaystyle\int_{-1/2}^{1/2}\frac{\cos^{-1}x}{e^{-x^2}}\,dx$

D. $\displaystyle\int_{-1/2}^{1/2}\frac{\tan^{-1}x}{e^x+e^{-x}}\,dx$

Find
$\int\frac{1}{\sqrt{5+4x-x^2}}\,dx.$

Using the substitution $t=\tan\frac{x}{2}$, find
$\int\frac{dx}{1+\cos x-\sin x}.$

Evaluate $\displaystyle\int \sin^3(2x)\cos(2x)\,dx$.

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

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

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

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

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

Find $\displaystyle\int\sin^{-1}(x)\,dx$.

Find $\displaystyle\int\frac{1}{2+\cos x}\,dx$.