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\includegraphics[max width=\textwidth, alt={}, center]{b2136f5d-0d66-4524-bb76-fcc4cb59150c-3_639_387_1749_879}
The diagram shows the curve \(y = \mathrm { e } ^ { 2 \sin x } \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
- Using the substitution \(u = \sin x\), find the exact value of the area of the shaded region bounded by the curve and the axes.
- Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.