12.4 Answers to 2012 test

2. For the partial fractions, we have $\frac{5x+4}{(x+3)(x^{2}+2)}=\frac{x+2}{x^{2}+2}-\frac{1}{x+3}$. For the indefinite integral, we have

The integral converges to $\frac{\pi\sqrt{2}}{2}+\frac{1}{2}\log\frac{9}{2}$.

3. a) $\frac{dy}{dx}=\frac{\sin t+t\cos t}{\cos t-t\sin t}$.

b) When $t=0$, we have $\frac{dy}{dx}=\frac{0}{1}=0$ so the tangent line is parallel to the $x$-axis.

c) The arc length is $\frac{1}{2}\sinh^{-1}\pi+\frac{1}{2}\pi\sqrt{1+\pi^{2}}=\frac{1}{2}\log(\pi+\sqrt{1+\pi^{2}})+\frac{1}{2}\pi\sqrt{1+\pi^{2}}$.

4. a) False, since the region of integration $a\leq y\leq b$, $c\leq x\leq d$ is different to the region of integration $a\leq x\leq b$, $c\leq y\leq d$.

b) True, since the Hessian discriminant $\Delta=-3$ is always negative.

c) False: the gradient is $-\frac{f_{x}}{f_{y}}$.

5. The value of the integral is $\frac{53}{6}$.