Use integration by parts to find \(\int x \ln x \mathrm {~d} x\).
Given that \(y = ( \ln x ) ^ { 2 }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
(2 marks)
The diagram shows part of the curve with equation \(y = \sqrt { x } \ln x\).
\includegraphics[max width=\textwidth, alt={}, center]{7148f43d-dc7d-43e2-b96e-ed1fb94073bf-5_406_645_696_719}
The shaded region \(R\) is bounded by the curve \(y = \sqrt { x } \ln x\), the line \(x = \mathrm { e }\) and the \(x\)-axis from \(x = 1\) to \(x = \mathrm { e }\).
Find the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis, giving your answer in an exact form.
(6 marks)