\(\quad y = \mathrm { e } ^ { - x } ( \sin x + \cos x )\)
Find \(\frac { d y } { d x }\) and simplify your answer.
Hence, show that
$$\int e ^ { - x } \sin x d x = a e ^ { - x } ( \sin x + \cos x ) + c$$
where \(a\) is a rational number.
A sketch of the graph of \(y = \mathrm { e } ^ { - x } \sin x\) for \(x \geq 0\) is shown below.
The areas of the finite regions bounded by the curve and the \(x\)-axis are denoted by \(A _ { 1 }\), \(A _ { 2 } , \ldots , A _ { n } , \ldots\)
\includegraphics[max width=\textwidth, alt={}, center]{bc7fb499-9462-40ae-88f4-87fc60f6a005-34_807_1246_959_406}
Find the exact value of the area \(A _ { 1 }\)
Show that
$$\frac { A _ { 2 } } { A _ { 1 } } = e ^ { - \pi }$$
Given that
$$\frac { A _ { n + 1 } } { A _ { n } } = e ^ { - \pi }$$
show that the exact value of the total area enclosed between the curve and the \(x\)-axis is
$$\frac { 1 + e ^ { \pi } } { 2 \left( e ^ { \pi } - 1 \right) }$$