| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Improper integrals with infinite upper limit (power/logarithm functions) |
| Difficulty | Standard +0.3 This is a straightforward improper integrals question requiring standard technique: evaluate the antiderivative and take the limit as the upper bound approaches infinity. Part (a) uses basic power rule (x^{-1/2}) and part (b) uses x^{-3/2}. Both integrals converge, requiring students to recognize when 1/∞ = 0. This is slightly easier than average since it's direct application of a well-practiced technique with no algebraic manipulation or partial fractions despite the topic label. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums4.08c Improper integrals: infinite limits or discontinuous integrands |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\int x^{-1/2} dx = 2x^{1/2} (+c)\); \(x^{1/2} \to \infty\) as \(x \to \infty\), so no value | M1A1, E1 | 3 marks |
| (b) \(\int x^{-3/2} dx = -2x^{-1/2} (+c)\); \(x^{-1/2} \to 0\) as \(x \to \infty\); \(\int_9^{\infty} x^{-3/2} dx = 2(0 - \frac{1}{3}) = \frac{2}{3}\) | M1A1, E1, A1 | 4 marks |
**(a)** $\int x^{-1/2} dx = 2x^{1/2} (+c)$; $x^{1/2} \to \infty$ as $x \to \infty$, so no value | M1A1, E1 | 3 marks | M1 for correct power in integral
**(b)** $\int x^{-3/2} dx = -2x^{-1/2} (+c)$; $x^{-1/2} \to 0$ as $x \to \infty$; $\int_9^{\infty} x^{-3/2} dx = 2(0 - \frac{1}{3}) = \frac{2}{3}$ | M1A1, E1, A1 | 4 marks | M1 for correct power in integral; PI; Allow A1 for correct answer even if not fully explained
**Total: 7 marks**3 For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
\begin{enumerate}[label=(\alph*)]
\item $\int _ { 9 } ^ { \infty } \frac { 1 } { \sqrt { x } } \mathrm {~d} x$;
\item $\int _ { 9 } ^ { \infty } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2008 Q3 [7]}}