\begin{homework}\
\label{homework:13}

\medskip
\exercise
Find the anti-derivative of: 

(a) $x\sin(ax+b)$; 
(b) $x^2\cos x$; 
(c) $\arctan\,x$; 
(d) $x^3e^x$; 
(e) $x\,\arctan\,x$.

\smallskip
\exercise
Find the center of mass of the uniform flat plate bounded by 

(a) the $x$- and $y$-axes and the curve $e^{kx}$ in the second quadrant
(out to $-\infty$); 

(b) the curves $y=-x^2\ln x$ and $y=x^2\ln x$; 

(c) the curves $y=-x^n\ln x$ and $y=x^n\ln x$ (express your answer in terms
of $n$); 

(d) the curves $y=\sin^2(\pi x)$ and $y=-\sin^2(\pi x)$ to the left of
$x=1/4$.

\smallskip
\exercise
Find the volume of the container obtained by rotating $x=y\sin y$ around
the $y$-axis between $y=0$ and $y=\pi/3$.

\end{homework}

%---------------------------------------
