5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4de08317-5fb9-4789-8d57-ccf463224c78-14_614_858_303_552}
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\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of part of the curve with equation \(y = 4 x - x \mathrm { e } ^ { \frac { 1 } { 2 } x } , x \geqslant 0\)
The curve meets the \(x\)-axis at the origin \(O\) and cuts the \(x\)-axis at the point \(A\) .
- Find,in terms of \(\ln 2\) ,the \(x\) coordinate of the point \(A\) .
- Find \(\int x \mathrm { e } ^ { \frac { 1 } { 2 } x } \mathrm {~d} x\)
The finite region \(R\) ,shown shaded in Figure 2,is bounded by the \(x\)-axis and the curve with equation \(y = 4 x - x \mathrm { e } ^ { \frac { 1 } { 2 } x } , x \geqslant 0\)
- Find,by integration,the exact value for the area of \(R\) .
Give your answer in terms of \(\ln 2\)
\includegraphics[max width=\textwidth, alt={}, center]{4de08317-5fb9-4789-8d57-ccf463224c78-18_2655_1943_114_118}