| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2013 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Improper integrals with discontinuity |
| Difficulty | Challenging +1.2 This is a Further Maths FP3 question requiring recognition of improper integrals and integration by parts with a limit. Part (a) requires identifying that ln(x) → -∞ as x → 0⁺. Part (b) is a standard application of integration by parts followed by evaluating a limit, though students must correctly handle the limiting behavior of x⁵ln(x) as x → 0⁺. While methodical, this requires more sophistication than typical A-level integration and involves careful limit evaluation, placing it moderately above average difficulty. |
| Spec | 1.08i Integration by parts4.08c Improper integrals: infinite limits or discontinuous integrands |
4
\begin{enumerate}[label=(\alph*)]
\item Explain why $\int _ { 0 } ^ { 1 } x ^ { 4 } \ln x \mathrm {~d} x$ is an improper integral.\\
(l mark)
\item Evaluate $\int _ { 0 } ^ { 1 } x ^ { 4 } \ln x \mathrm {~d} x$, showing the limiting process used.\\
(6 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2013 Q4 [7]}}