| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2011 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Improper integrals with partial fractions (infinite limit) |
| Difficulty | Standard +0.8 This is a Further Maths question combining partial fractions with improper integrals. Part (a) is routine algebraic manipulation (reverse partial fractions). Part (b) requires setting up a limit, integrating logarithmic terms, applying logarithm laws, and evaluating the limit as x→∞. While methodical, it demands careful handling of multiple techniques and the improper integral concept, placing it moderately above average difficulty. |
| Spec | 1.02y Partial fractions: decompose rational functions4.08c Improper integrals: infinite limits or discontinuous integrands |
5
\begin{enumerate}[label=(\alph*)]
\item Write $\frac { 4 } { 4 x + 1 } - \frac { 3 } { 3 x + 2 }$ in the form $\frac { C } { ( 4 x + 1 ) ( 3 x + 2 ) }$, where $C$ is a constant.\\
(l mark)
\item Evaluate the improper integral
$$\int _ { 1 } ^ { \infty } \frac { 10 } { ( 4 x + 1 ) ( 3 x + 2 ) } d x$$
showing the limiting process used and giving your answer in the form $\ln k$, where $k$ is a constant.\\
(6 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2011 Q5 [7]}}