| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 5 |
| 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 integral question requiring knowledge of the convergence test for power functions (integral of x^p converges for p < -1) and routine integration of power functions. While improper integrals are a Further Maths topic, this particular question involves direct application of a standard result with minimal calculation, making it slightly easier than average. |
| Spec | 4.08c Improper integrals: infinite limits or discontinuous integrands |
| Answer | Marks | Guidance |
|---|---|---|
| Part | Answer/Working | Mark |
| (a) | \(\int x^{-3/2} dx = 3x^{1/2} (+c)\) | B1 |
| \((3)x^{1/2} \to \infty\) as \(x \to \infty\), so no finite value | E1 | |
| (b) | \(\int x^{-1/2} dx = -3x^{-1/2} (+c)\) | M1 |
| A1 | \(-3x^{-1/2}\) OE | |
| \(\int_8^x x^{-1/2} dx = -3(0 - \frac{1}{2}) = \frac{3}{2}\) | A1 | 5 marks total; CSO |
| Part | Answer/Working | Mark | Guidance |
|------|---|---|---|
| (a) | $\int x^{-3/2} dx = 3x^{1/2} (+c)$ | B1 | $kx^{1/2}$, $k \neq 0$ ie condone incorrect non-zero coefficient here |
| | $(3)x^{1/2} \to \infty$ as $x \to \infty$, so no finite value | E1 | |
| (b) | $\int x^{-1/2} dx = -3x^{-1/2} (+c)$ | M1 | $\lambda x^{-1/2}$, $\lambda \neq 0$ |
| | | A1 | $-3x^{-1/2}$ OE |
| | $\int_8^x x^{-1/2} dx = -3(0 - \frac{1}{2}) = \frac{3}{2}$ | A1 | 5 marks total; CSO |2 Show that only one of the following improper integrals has a finite value, and find that value:
\begin{enumerate}[label=(\alph*)]
\item $\quad \int _ { 8 } ^ { \infty } x ^ { - \frac { 2 } { 3 } } \mathrm {~d} x$;
\item $\quad \int _ { 8 } ^ { \infty } x ^ { - \frac { 4 } { 3 } } \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2012 Q2 [5]}}