| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2011 |
| 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 integration by parts with ln x (standard technique), recognition of an improper integral at x=0 (conceptual understanding), and evaluation using limits. While methodical, it combines multiple concepts and requires careful handling of the limit as x→0⁺ for x²ln x, placing it moderately above average difficulty. |
| Spec | 1.08i Integration by parts4.08c Improper integrals: infinite limits or discontinuous integrands |
3
\begin{enumerate}[label=(\alph*)]
\item Find $\int x ^ { 2 } \ln x \mathrm {~d} x$.
\item Explain why $\int _ { 0 } ^ { \mathrm { e } } x ^ { 2 } \ln x \mathrm {~d} x$ is an improper integral.
\item Evaluate $\int _ { 0 } ^ { \mathrm { e } } x ^ { 2 } \ln x \mathrm {~d} x$, showing the limiting process used.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2011 Q3 [7]}}