| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Reduction Formulae |
| Type | Hyperbolic function reduction |
| Difficulty | Challenging +1.8 This is a Further Maths FP3 reduction formula question involving hyperbolic functions. Part (a) requires product rule differentiation with hyperbolic identities (standard technique). Part (b) involves integration by parts using the given result to establish a recurrence relation. Part (c) requires iterative application of the reduction formula with careful evaluation of base cases. While the techniques are standard for FP3, the hyperbolic context, multi-step reasoning, and need to track multiple iterations elevate this above average difficulty, though it remains a recognizable textbook-style reduction formula problem. |
| Spec | 4.06b Method of differences: telescoping series4.07d Differentiate/integrate: hyperbolic functions |
Given that $y = \sinh^{n-1} x \cosh x$,
\begin{enumerate}[label=(\alph*)]
\item show that $\frac{dy}{dx} = (n-1) \sinh^{n-2} x + n \sinh^n x$. [3]
\end{enumerate}
The integral $I_n$ is defined by $I_n = \int_0^{\operatorname{arsinh} 1} \sinh^n x \, dx$, $n \geq 0$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Using the result in part (a), or otherwise, show that
$$nI_n = \sqrt{2} - (n-1)I_{n-2}, \quad n \geq 2$$ [2]
\item Hence find the value of $I_4$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q24 [9]}}