3. (a) Use the substitution \(u = 2 - x ^ { 2 }\) to find
$$\int \frac { x } { 2 - x ^ { 2 } } \mathrm {~d} x$$
(b) Evaluate
$$\int _ { 0 } ^ { \frac { \pi } { 4 } } \sin 3 x \cos x d x$$
- continued
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{beeaedf6-62e8-4649-b023-1b7e2be9957e-06_636_837_146_511}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows the curve with equation \(y = x \sqrt { \ln x } , x \geq 1\).
The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 3\).
(a) Using the trapezium rule with two intervals of equal width, estimate the area of the shaded region.
The shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
(b) Find the exact volume of the solid formed.