CAIE P3 2016 June — Question 2 5 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2016
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration by Parts
TypeBasic integration by parts
DifficultyModerate -0.8 This is a straightforward single application of integration by parts with simple functions (polynomial times exponential) and standard limits. It requires only routine technique with no problem-solving insight, making it easier than average but not trivial since students must correctly apply the formula and evaluate at the given bounds.
Spec1.08i Integration by parts
2 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\).
Question 2:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Integrate by parts and reach \(axe^{-2x} + b\int e^{-2x}\,dx\)M1
Obtain \(-\frac{1}{2}xe^{-2x} + \frac{1}{2}\int e^{-2x}\,dx\), or equivalentA1
Complete integration correctly, obtaining \(-\frac{1}{2}xe^{-2x} - \frac{1}{4}e^{-2x}\), or equivalentA1
Use limits \(x = 0\) and \(x = \frac{1}{2}\) correctly, having integrated twiceM1
Obtain \(\frac{1}{4} - \frac{1}{2}e^{-1}\), or exact equivalentA1
Total[5] 
## Question 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Integrate by parts and reach $axe^{-2x} + b\int e^{-2x}\,dx$ | M1 | |
| Obtain $-\frac{1}{2}xe^{-2x} + \frac{1}{2}\int e^{-2x}\,dx$, or equivalent | A1 | |
| Complete integration correctly, obtaining $-\frac{1}{2}xe^{-2x} - \frac{1}{4}e^{-2x}$, or equivalent | A1 | |
| Use limits $x = 0$ and $x = \frac{1}{2}$ correctly, having integrated twice | M1 | |
| Obtain $\frac{1}{4} - \frac{1}{2}e^{-1}$, or exact equivalent | A1 | |
| **Total** | **[5]** | |

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2 Find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 2 } } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x$.

\hfill \mbox{\textit{CAIE P3 2016 Q2 [5]}}
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Q1 5 Q2 5 Q3 5 Q4 6 Q5 6 Q6 7 Q7 9 Q8 10 Q9 11 Q10 11