4. (a) Find $\frac { d y } { d x }$ for each of the functions $$\begin{aligned}
& y = \sin ( \ln x ) \\
& y = x \sin ( \ln x ) \\
& y = x \cos ( \ln x )
\end{aligned}$$ (b) Sketch the following curves using the axes provided on the next page: (i) $y = \ln x$, for $1 \leqslant x \leqslant e ^ { \pi }$, (ii) $y = \sin ( \ln x )$, for $1 \leqslant x \leqslant e ^ { \pi }$. (c) Evaluate $$\int _ { 1 } ^ { e ^ { \pi } } \sin ( \ln x ) d x$$ [Figure]
4. (a) Find $\frac { d y } { d x }$ for each of the functions
$$\begin{aligned}
& y = \sin ( \ln x ) \\
& y = x \sin ( \ln x ) \\
& y = x \cos ( \ln x )
\end{aligned}$$
(b) Sketch the following curves using the axes provided on the next page:\\
(i) $y = \ln x$, for $1 \leqslant x \leqslant e ^ { \pi }$,\\
(ii) $y = \sin ( \ln x )$, for $1 \leqslant x \leqslant e ^ { \pi }$.\\
(c) Evaluate
$$\int _ { 1 } ^ { e ^ { \pi } } \sin ( \ln x ) d x$$
\includegraphics[max width=\textwidth, alt={}, center]{b0fa88de-4f77-455b-bdf5-5df98552db9c-09_1155_1230_148_319}\\