| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Reduction Formulae |
| Type | Exponential times polynomial |
| Difficulty | Challenging +1.2 This is a standard reduction formula question requiring integration by parts to establish the recurrence relation, then applying it twice with straightforward arithmetic. While it involves Further Maths content (FP3), the technique is mechanical and follows a well-practiced pattern with no novel insight required. The multi-step calculation and algebraic manipulation place it moderately above average difficulty. |
| Spec | 1.08i Integration by parts4.06b Method of differences: telescoping series |
$$I_n = \int_0^1 x^n e^{2x} \, dx, \quad n \geq 0.$$
\begin{enumerate}[label=(\alph*)]
\item Prove that, for $n \geq 1$,
$$I_n = \frac{1}{2}(x^n e^{2x} - nI_{n-1}).$$ [3]
\item Find, in terms of $e$, the exact value of
$$\int_0^1 x^2 e^{2x} \, dx.$$ [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q32 [8]}}