7.

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
  \includegraphics[alt={},max width=\textwidth]{25f0c7cc-0701-4836-931e-0eff5145e029-4_446_1131_1093_567}
\end{center}
\end{figure}

Figure 3 shows a sketch of part of the curve $C$ with question

$$y = \mathrm { e } ^ { - x } \sin x , \quad x \geq 0 .$$
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of the points $P , Q$ and $R$ where $C$ cuts the positive axis.
\item 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.
\item Find an expression for $A _ { n }$ in terms of $n$ and $\pi$ .\\
(6)
\item Show that $A _ { 1 } + A _ { 2 } + \ldots + A _ { n } + \ldots$ is a geometric series with sum to infinity

$$\frac { \mathrm { e } ^ { \pi } } { 2 \left( \mathrm { e } ^ { \pi } - 1 \right) } .$$
\item 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
\end{enumerate}

\hfill \mbox{\textit{Edexcel AEA 2003 Q7 [22]}}