| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Improper integral with parts |
| Difficulty | Standard +0.8 This is a Further Maths question requiring two applications of integration by parts for part (a), then careful handling of an improper integral with limits approaching infinity in part (b). While the technique is standard for FP3, the combination of repeated integration by parts plus rigorous treatment of the limiting process elevates this above typical A-level questions. |
| Spec | 1.08i Integration by parts4.08c Improper integrals: infinite limits or discontinuous integrands |
5
\begin{enumerate}[label=(\alph*)]
\item Find $\int x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x$.
\item Hence evaluate $\int _ { 0 } ^ { \infty } x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x$, showing the limiting process used.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2012 Q5 [7]}}