| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Reduction Formulae |
| Type | Exponential times polynomial |
| Difficulty | Standard +0.8 This is a standard FP3 reduction formula question requiring integration by parts multiple times and algebraic manipulation. While it involves several steps and careful bookkeeping across five parts, the techniques are routine for Further Maths students: deriving reduction formulae by parts, applying them iteratively, and using hyperbolic function definitions. The multi-part structure guides students through the solution systematically, making it moderately challenging but not requiring novel insight. |
| Spec | 1.08i Integration by parts4.06b Method of differences: telescoping series4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials |
$$I_n = \int_0^1 x^n e^x \, dx \text{ and } J_n = \int_0^1 x^n e^{-x} \, dx, \quad n \geq 0.$$
\begin{enumerate}[label=(\alph*)]
\item Show that, for $n \geq 1$,
$$I_n = e - nI_{n-1}.$$ [2]
\item Find a similar reduction formula for $J_n$. [3]
\item Show that $J_2 = 2 - \frac{5}{e}$. [3]
\item Show that $\int_0^1 x^n \cosh x \, dx = \frac{1}{2}(I_n + J_n)$. [1]
\item Hence, or otherwise, evaluate $\int_0^1 x^2 \cosh x \, dx$, giving your answer in terms of $e$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q15 [13]}}