AQA FP1 2013 January — Question 4 4 marks

Exam BoardAQA
ModuleFP1 (Further Pure Mathematics 1)
Year2013
SessionJanuary
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration with Partial Fractions
TypeImproper integrals with infinite upper limit (power/logarithm functions)
DifficultyStandard +0.3 This is a straightforward improper integral requiring recognition that 1/(x√x) = x^(-3/2), integration using the power rule to get -2x^(-1/2), and evaluating the limit as the upper bound approaches infinity. The limit evaluation is simple since x^(-1/2) → 0. While it's a Further Maths topic, it's a standard textbook exercise with no conceptual challenges beyond basic technique application.
Spec4.08c Improper integrals: infinite limits or discontinuous integrands
4 Show that the improper integral \(\int _ { 25 } ^ { \infty } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x\) has a finite value and find that value.
Question 4:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\int \frac{1}{x\sqrt{x}}\,dx = \int x^{-\frac{3}{2}}\,dx\)M1 \(\int x^{-\frac{3}{2}}\) PI
\(= -2x^{-\frac{1}{2}} (+c)\)A1 ACF, condone absence of \(+c\)
\(-2x^{-\frac{1}{2}} \to 0\) as \(x \to \infty\)E1 OE. Ft on \(kx^{-n}\), \(n>0\)
\(\int_{25}^{\infty} \frac{1}{x\sqrt{x}}\,dx = \frac{2}{5}\)A1 
## Question 4:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int \frac{1}{x\sqrt{x}}\,dx = \int x^{-\frac{3}{2}}\,dx$ | M1 | $\int x^{-\frac{3}{2}}$ PI |
| $= -2x^{-\frac{1}{2}} (+c)$ | A1 | ACF, condone absence of $+c$ |
| $-2x^{-\frac{1}{2}} \to 0$ as $x \to \infty$ | E1 | OE. Ft on $kx^{-n}$, $n>0$ |
| $\int_{25}^{\infty} \frac{1}{x\sqrt{x}}\,dx = \frac{2}{5}$ | A1 | |

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4 Show that the improper integral $\int _ { 25 } ^ { \infty } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x$ has a finite value and find that value.

\hfill \mbox{\textit{AQA FP1 2013 Q4 [4]}}
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