| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Improper integrals with discontinuity |
| Difficulty | Challenging +1.2 This is a Further Maths question testing understanding of improper integrals with discontinuities at endpoints. Part (a) requires recognizing that x^(-1/3) has an integrable singularity at x=0, while part (b) requires identifying that the integrand simplifies to x^(-2/3) + x^(-4/3), where x^(-4/3) diverges. The conceptual understanding needed (convergence vs divergence of power functions) elevates this above routine integration, but the actual calculations are straightforward once the concept is grasped. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums4.08c Improper integrals: infinite limits or discontinuous integrands |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\int \left(x^{\frac{1}{3}} + x^{-\frac{1}{3}}\right) dx = \frac{3}{4}x^{\frac{4}{3}} + \frac{3}{2}x^{\frac{2}{3}} (+c)\) | M1A1 | M1 for adding 1 to index at least once |
| \(\int \ldots = \left(\frac{3}{4} + \frac{3}{2}\right) - 0 = \frac{9}{4}\) | m1A1 | Condone no mention of limiting process; m1 if "\(-0\)" stated or implied |
| Answer | Marks | Guidance |
|---|---|---|
| (b) Second term is \(x^{-\frac{4}{3}}\) | B1 | |
| Integral of this is \(-3x^{-\frac{1}{3}}\) | M1A1 | M1 for correct index |
| \(x^{-\frac{1}{3}} \to \infty\) as \(x \to 0\), so no value | E1 |
**(a)** $\int \left(x^{\frac{1}{3}} + x^{-\frac{1}{3}}\right) dx = \frac{3}{4}x^{\frac{4}{3}} + \frac{3}{2}x^{\frac{2}{3}} (+c)$ | M1A1 | M1 for adding 1 to index at least once
$\int \ldots = \left(\frac{3}{4} + \frac{3}{2}\right) - 0 = \frac{9}{4}$ | m1A1 | Condone no mention of limiting process; m1 if "$-0$" stated or implied
**Total: 4 marks**
**(b)** Second term is $x^{-\frac{4}{3}}$ | B1 |
Integral of this is $-3x^{-\frac{1}{3}}$ | M1A1 | M1 for correct index
$x^{-\frac{1}{3}} \to \infty$ as $x \to 0$, so no value | E1 |
**Total: 4 marks**
---8 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 $\quad \int _ { 0 } ^ { 1 } \left( x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 1 } { 3 } } \right) \mathrm { d } x$;
\item $\int _ { 0 } ^ { 1 } \frac { x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 1 } { 3 } } } { x } \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2007 Q8 [8]}}