Given that \(y = x ^ { - 2 } \ln x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 - 2 \ln x } { x ^ { 3 } }\).
Using integration by parts, find \(\int x ^ { - 2 } \ln x \mathrm {~d} x\).
The sketch shows the graph of \(y = x ^ { - 2 } \ln x\).
\includegraphics[max width=\textwidth, alt={}, center]{9aac4ee4-2435-4315-a87d-fe9fa8e15665-007_593_1034_696_543}
Using the answer to part (a), find, in terms of e, the \(x\)-coordinate of the stationary point \(A\).
The region \(R\) is bounded by the curve, the \(x\)-axis and the line \(x = 5\). Using your answer to part (b), show that the area of \(R\) is
$$\frac { 1 } { 5 } ( 4 - \ln 5 )$$