CAIE P3 2017 June — Question 4 4 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2017
SessionJune
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration by Parts
TypeBasic integration by parts
DifficultyModerate -0.3 This is a straightforward single application of integration by parts with standard functions (polynomial times trigonometric). The limits are clean (0 and π/2), and the substitution u=θ, dv=sin(½θ)dθ is routine. It requires careful execution but no problem-solving insight, making it slightly easier than average.
Spec1.08i Integration by parts
4 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \theta \sin \frac { 1 } { 2 } \theta \mathrm {~d} \theta\).
Question 4:
AnswerMarks Guidance
AnswerMark Guidance
Integrate by parts and reach \(a\theta\cos\frac{1}{2}\theta + b\int\cos\frac{1}{2}\theta\, d\theta\)*M1
Complete integration and obtain indefinite integral \(-2\theta\cos\frac{1}{2}\theta + 4\sin\frac{1}{2}\theta\)A1
Substitute limits correctly, having integrated twiceDM1
Obtain final answer \(\frac{(4-\pi)}{\sqrt{2}}\), or exact equivalentA1
Total: 4 
## Question 4:
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate by parts and reach $a\theta\cos\frac{1}{2}\theta + b\int\cos\frac{1}{2}\theta\, d\theta$ | *M1 | |
| Complete integration and obtain indefinite integral $-2\theta\cos\frac{1}{2}\theta + 4\sin\frac{1}{2}\theta$ | A1 | |
| Substitute limits correctly, having integrated twice | DM1 | |
| Obtain final answer $\frac{(4-\pi)}{\sqrt{2}}$, or exact equivalent | A1 | |
| **Total: 4** | | |

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4 Find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \theta \sin \frac { 1 } { 2 } \theta \mathrm {~d} \theta$.\\

\hfill \mbox{\textit{CAIE P3 2017 Q4 [4]}}
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