3.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{89d4a7a5-3f4f-4d16-b14e-a27243cedd78-05_620_867_301_536}
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\caption{Figure 1}
\end{figure}
Figure 1 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 1, 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 \(\ln 2\)