$[9233/$N$2003/$II$/3]$

$($a$)$ Without the use of a calculator, evaluate ${\displaystyle \underset{1}{\overset{3}{}}}|x(2-x)|dx.$ By sketching the graphs of $y=e^x$ and $y=e^{-x},$ or otherwise, solve the inequality

$($b$)e^x-e^{-x}>0.$ Hence, find the exact value of ${\displaystyle \underset{ln}{\overset{1}{}}}(\frac{?}{2})|e^x-e^{-x}|dx.$

Ans: $($a$)2$; $($b$)$e $+\frac{1}{2}-\frac{3}{2}$