| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Improper integrals with discontinuity |
| Difficulty | Standard +0.8 This FP1 question tests understanding of improper integrals with discontinuities at endpoints, requiring students to recognize when integrals converge vs diverge and explain the concept. While the integration itself is straightforward (basic power rule), the conceptual understanding of improper integrals and the comparison between convergent/divergent cases elevates this above routine A-level questions. The explanation component adds depth beyond mechanical calculation. |
| Spec | 4.08c Improper integrals: infinite limits or discontinuous integrands |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) \(\int x^{-\frac{1}{2}} dx = 2x^{\frac{1}{2}} (+ c)\) | M1A1 | M1 for \(kx^{-\frac{1}{2}}\) |
| \(\int_0^9 \frac{1}{\sqrt{x}} dx = 6\) | A1√ | ft wrong coeff of \(x^{\frac{1}{2}}\) |
| (ii) \(\int x^{-\frac{1}{2}} dx = -2x^{-\frac{1}{2}} (+ c)\) | M1A1 | M1 for \(kx^{-\frac{1}{2}}\) |
| \(x^{-\frac{1}{2}} \to \infty\) as \(x \to 0\), so no value | E1; E1 | 'Tending to infinity' clearly implied |
**(a)(i)** $\int x^{-\frac{1}{2}} dx = 2x^{\frac{1}{2}} (+ c)$ | M1A1 | M1 for $kx^{-\frac{1}{2}}$
$\int_0^9 \frac{1}{\sqrt{x}} dx = 6$ | A1√ | ft wrong coeff of $x^{\frac{1}{2}}$
**(ii)** $\int x^{-\frac{1}{2}} dx = -2x^{-\frac{1}{2}} (+ c)$ | M1A1 | M1 for $kx^{-\frac{1}{2}}$
$x^{-\frac{1}{2}} \to \infty$ as $x \to 0$, so no value | E1; E1 | 'Tending to infinity' clearly implied
**Total for Q2: 7 marks**
---2
\begin{enumerate}[label=(\alph*)]
\item 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=(\roman*)]
\item $\int _ { 0 } ^ { 9 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x$;
\item $\int _ { 0 } ^ { 9 } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x$.
\end{enumerate}\item Explain briefly why the integrals in part (a) are improper integrals.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2006 Q2 [7]}}