12.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{48f9a252-61a2-491d-94d0-8470aee96942-18_1038_1271_244_440}
\captionsetup{labelformat=empty}
\caption{Figure 6}
\end{figure}
The figure 6 shows a sketch of the curve with equation
$$y = x ^ { 2 } \ln 2 x$$
The finite region \(R\), shown shaded in figure 5, is bounded by the line with equation \(x = \frac { e ^ { 2 } } { 2 }\), the curve \(C\), the line with equation \(x = e ^ { 2 }\) and the \(x\)-axis.
Show that the exact value of the area of region \(R\) is \(\frac { e ^ { 6 } } { 72 } ( 35 + 24 \ln 2 )\).