9758/2021/P1/Q09

A function $\mathrm{f}$ is defined by $\mathrm{f}(x)=\mathrm{e}^x \cos x$, for $0 \leqslant x \leqslant \frac{1}{2} \pi$.
(a) Using calculus, find the stationary point of $\mathrm{f}(x)$ and determine its nature.

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(b) Integrate by parts twice to show that
$\int \mathrm{e}^{2 x} \cos 2 x \mathrm{~d} x=\frac{1}{4} \mathrm{e}^{2 x}(\sin 2 x+\cos 2 x)+c$

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(c) The graph of $y=\mathrm{f}(x)$ is rotated completely about the $x$-axis. Find the exact volume generated.

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