Standard +0.8 This is a well-structured multi-part question requiring product rule differentiation, recognition of an integration result, definite integration with exponential-trigonometric functions, and geometric series summation. While each individual step uses standard techniques, the question requires connecting multiple concepts (differentiation→integration, areas→geometric progression→infinite series) and careful algebraic manipulation. The geometric series application to infinite area is slightly beyond routine A-level questions but remains accessible to strong Further Maths students.
\(\quad y = \mathrm { e } ^ { - x } ( \sin x + \cos x )\)
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
$$\frac { 1 + e ^ { \pi } } { 2 \left( e ^ { \pi } - 1 \right) }$$
14.
\begin{enumerate}[label=(\alph*)]
\item $\quad y = \mathrm { e } ^ { - x } ( \sin x + \cos x )$
Find $\frac { d y } { d x }$ and simplify your answer.
\item Hence, show that
$$\int e ^ { - x } \sin x d x = a e ^ { - x } ( \sin x + \cos x ) + c$$
where $a$ is a rational number.
\item 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}
\begin{enumerate}[label=(\roman*)]
\item Find the exact value of the area $A _ { 1 }$
\item Show that
$$\frac { A _ { 2 } } { A _ { 1 } } = e ^ { - \pi }$$
\item 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
$$\frac { 1 + e ^ { \pi } } { 2 \left( e ^ { \pi } - 1 \right) }$$
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2025 Q14 [12]}}