| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2012 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Improper integrals with infinite upper limit (exponential/IBP) |
| Difficulty | Challenging +1.3 This is a structured Further Maths question where the substitution is given explicitly, making part (b) a careful but guided calculation. Part (c) requires evaluating limits as x→∞, which is standard for FP3 improper integrals. While it involves multiple techniques (substitution, improper integrals, limits), the scaffolding and routine nature of following given instructions places it moderately above average difficulty. |
| Spec | 1.08h Integration by substitution4.08c Improper integrals: infinite limits or discontinuous integrands |
5
\begin{enumerate}[label=(\alph*)]
\item Explain why $\int _ { \frac { 1 } { 2 } } ^ { \infty } \frac { x ( 1 - 2 x ) } { x ^ { 2 } + 3 \mathrm { e } ^ { 4 x } } \mathrm {~d} x$ is an improper integral.\\
(1 mark)
\item By using the substitution $u = x ^ { 2 } \mathrm { e } ^ { - 4 x } + 3$, find
$$\int \frac { x ( 1 - 2 x ) } { x ^ { 2 } + 3 \mathrm { e } ^ { 4 x } } \mathrm {~d} x$$
\item Hence evaluate $\int _ { \frac { 1 } { 2 } } ^ { \infty } \frac { x ( 1 - 2 x ) } { x ^ { 2 } + 3 \mathrm { e } ^ { 4 x } } \mathrm {~d} x$, showing the limiting process used.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2012 Q5 [8]}}