OCR C4 2011 June — Question 4 6 marks

Exam BoardOCR
ModuleC4 (Core Mathematics 4)
Year2011
SessionJune
Marks6
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TopicIntegration by Substitution
TypeTrigonometric substitution: direct evaluation
DifficultyStandard +0.8 This is a moderately challenging C4 integration question requiring trigonometric substitution with a fractional coefficient, manipulation of trigonometric identities (1-9x² becomes cos²θ), careful handling of the power 3/2, and conversion of limits. While the substitution is given, students must execute multiple steps correctly including dx/dθ, simplifying the resulting trigonometric expression, and finding exact values at non-standard angles (θ = π/6).
Spec1.08h Integration by substitution
4 Use the substitution \(x = \frac { 1 } { 3 } \sin \theta\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 6 } } \frac { 1 } { \left( 1 - 9 x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$
Question 4:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Indefinite integral: attempt to connect \(dx\) and \(d\theta\)M1 Incl \(\frac{dx}{d\theta}=,\frac{d\theta}{dx}=\) , \(dx=\ldots d\theta\); not \(dx=d\theta\)
Denominator \((1-9x^2)^{3/2}\) becomes \(\cos^3\theta\)B1
Reduce original integral to \(\frac{1}{3}\int\frac{1}{\cos^2\theta}\,d\theta\)A1 May be implied, seen only as \(\frac{1}{3}\int\sec^2\theta\,d\theta\)
Change \(\int\frac{1}{\cos^2\theta}\,d\theta\) to \(\tan\theta\)B1 Ignore \(\frac{1}{3}\) at this stage
Use appropriate limits for \(\theta\) (allow degrees) or \(x\)M1 Integration need not be accurate
\(\frac{\sqrt{3}}{9}\) AEF, exact answer required, ISWA1 6 
# Question 4:

| Answer/Working | Marks | Guidance |
|---|---|---|
| Indefinite integral: attempt to connect $dx$ and $d\theta$ | M1 | Incl $\frac{dx}{d\theta}=,\frac{d\theta}{dx}=$ , $dx=\ldots d\theta$; not $dx=d\theta$ |
| Denominator $(1-9x^2)^{3/2}$ becomes $\cos^3\theta$ | B1 | |
| Reduce original integral to $\frac{1}{3}\int\frac{1}{\cos^2\theta}\,d\theta$ | A1 | May be implied, seen only as $\frac{1}{3}\int\sec^2\theta\,d\theta$ |
| Change $\int\frac{1}{\cos^2\theta}\,d\theta$ to $\tan\theta$ | B1 | Ignore $\frac{1}{3}$ at this stage |
| Use appropriate limits for $\theta$ (allow degrees) or $x$ | M1 | Integration need not be accurate |
| $\frac{\sqrt{3}}{9}$ AEF, exact answer required, ISW | A1 **6** | |

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4 Use the substitution $x = \frac { 1 } { 3 } \sin \theta$ to find the exact value of

$$\int _ { 0 } ^ { \frac { 1 } { 6 } } \frac { 1 } { \left( 1 - 9 x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$

\hfill \mbox{\textit{OCR C4 2011 Q4 [6]}}
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