\(\quad y = \mathrm { e } ^ { - x } ( \sin x + \cos x )\)
Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
Simplify your answer.
16
Hence, show that
$$\int \mathrm { e } ^ { - x } \sin x \mathrm {~d} x = a \mathrm { e } ^ { - x } ( \sin x + \cos x ) + c$$
where \(a\) is a rational number.
16
A sketch of the graph of \(y = \mathrm { e } ^ { - x } \sin x\) for \(x \geq 0\) is shown below. \(A _ { 1 } , A _ { 2 } , \ldots , A _ { n } , \ldots\)
The areas of the finite regions bounded by the curve and the \(x\)-axis are denoted by
\includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-27_974_1507_502_262}
16
Find the exact value of the area \(A _ { 1 }\)
16
(ii) Show that
$$\frac { A _ { 2 } } { A _ { 1 } } = \mathrm { e } ^ { - \pi }$$
16
(iii) Given that
$$\frac { A _ { n + 1 } } { A _ { n } } = \mathrm { e } ^ { - \pi }$$
show that the exact value of the total area enclosed between the curve and the \(x\)-axis is
$$\frac { 1 + \mathrm { e } ^ { \pi } } { 2 \left( \mathrm { e } ^ { \pi } - 1 \right) }$$
\includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-30_2488_1719_219_150}
\includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-31_2488_1719_219_150}
\includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-32_2496_1721_214_148}