CAIE P3 2021 November — Question 4 5 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2021
SessionNovember
Marks5
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 setup is routine (u=x, dv=sin(½x)dx), and evaluating the definite integral requires careful arithmetic with the given limits but no conceptual challenges. Slightly easier than average due to being a direct textbook-style question.
Spec1.08i Integration by parts
4 Find the exact value of \(\int _ { \frac { 1 } { 3 } \pi } ^ { \pi } x \sin \frac { 1 } { 2 } x \mathrm {~d} x\).
Question 4:
AnswerMarks Guidance
AnswerMark Guidance
Commence integration and reach \(ax\cos\frac{1}{2}x + b\int\cos\frac{1}{2}x\,dx\)\*M1
Obtain \(-2x\cos\frac{1}{2}x + 2\int\cos\frac{1}{2}x\,dx\)A1 OE
Complete integration obtaining \(-2x\cos\frac{1}{2}x + 4\sin\frac{1}{2}x\)A1 OE
Use limits correctly, having integrated twiceDM1
Obtain answer \(2 + \frac{\sqrt{3}}{3}\pi\), or exact equivalentA1 
## Question 4:

| Answer | Mark | Guidance |
|--------|------|----------|
| Commence integration and reach $ax\cos\frac{1}{2}x + b\int\cos\frac{1}{2}x\,dx$ | \*M1 | |
| Obtain $-2x\cos\frac{1}{2}x + 2\int\cos\frac{1}{2}x\,dx$ | A1 | OE |
| Complete integration obtaining $-2x\cos\frac{1}{2}x + 4\sin\frac{1}{2}x$ | A1 | OE |
| Use limits correctly, having integrated twice | DM1 | |
| Obtain answer $2 + \frac{\sqrt{3}}{3}\pi$, or exact equivalent | A1 | |

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4 Find the exact value of $\int _ { \frac { 1 } { 3 } \pi } ^ { \pi } x \sin \frac { 1 } { 2 } x \mathrm {~d} x$.\\

\hfill \mbox{\textit{CAIE P3 2021 Q4 [5]}}
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