7.
\begin{figure}[h]
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\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{25f0c7cc-0701-4836-931e-0eff5145e029-4_446_1131_1093_567}
\end{figure}
Figure 3 shows a sketch of part of the curve \(C\) with question
$$y = \mathrm { e } ^ { - x } \sin x , \quad x \geq 0 .$$
- Find the coordinates of the points \(P , Q\) and \(R\) where \(C\) cuts the positive axis.
- Use integration by parts to show that
$$\int \mathrm { e } ^ { - x } \sin x \mathrm {~d} x = - \frac { 1 } { 2 } \mathrm { e } ^ { - x } ( \sin x + \cos x ) + \text { constant }$$
The terms of the sequence \(A _ { 1 } , A _ { 2 } , \ldots , A _ { n } , \ldots\) represent areas between \(C\) and the \(x\)-axis for successive portions of \(C\) where \(y\) is positive.The area represented by \(A _ { 1 }\) and \(A _ { 2 }\) are shown in Figure 3.
- Find an expression for \(A _ { n }\) in terms of \(n\) and \(\pi\) .
(6) - Show that \(A _ { 1 } + A _ { 2 } + \ldots + A _ { n } + \ldots\) is a geometric series with sum to infinity
$$\frac { \mathrm { e } ^ { \pi } } { 2 \left( \mathrm { e } ^ { \pi } - 1 \right) } .$$
- Given that
$$\int _ { 0 } ^ { \infty } \mathrm { e } ^ { - x } \sin x \mathrm {~d} x = \frac { 1 } { 2 }$$
find the exact value of
$$\int _ { 0 } ^ { \infty } \left| e ^ { - x } \sin x \right| d x$$
and simplify your answer.
END