4. Use the substitution \(x = 2 \sin \theta\) to find the exact value of
$$\int _ { 0 } ^ { \sqrt { 3 } } \frac { 1 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$
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Question 4:
Answer Marks
Guidance
Answer/Working Mark
Guidance
\(x = 2\sin\theta \Rightarrow \frac{dx}{d\theta} = 2\cos\theta\) B1
Condone \(x' = 2\cos\theta\)
\(\int \frac{1}{(4-x^2)^{3/2}}dx = \int \frac{1}{(4-4\sin^2\theta)^{3/2}} 2\cos\theta \, d\theta\) M1
Attempt integral in \(\theta\) only using \(x=2\sin\theta\) and \(dx = \pm A\cos\theta \, d\theta\)
\(= \int \frac{1}{4}\sec^2\theta \, d\theta\) M1
Uses \(1-\sin^2\theta = \cos^2\theta\) and simplifies to \(\int C\sec^2\theta \, d\theta\)
\(= \frac{1}{4}\tan\theta\) dM1A1
dM1 dependent on previous M1: \(\int \sec^2\theta \to \tan\theta\); A1 for \(\frac{1}{4}\tan\theta\) (no \(+c\) required)
Uses limits \(0\) and \(\frac{\pi}{3}\) in integrated expression M1
Changes \(x\)-limits to \(\theta\)-limits of \(0\) and \(\frac{\pi}{3}\)
\(= \left[\frac{1}{4}\tan\theta\right]_0^{\pi/3} = \frac{\sqrt{3}}{4}\) A1
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## Question 4:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x = 2\sin\theta \Rightarrow \frac{dx}{d\theta} = 2\cos\theta$ | B1 | Condone $x' = 2\cos\theta$ |
| $\int \frac{1}{(4-x^2)^{3/2}}dx = \int \frac{1}{(4-4\sin^2\theta)^{3/2}} 2\cos\theta \, d\theta$ | M1 | Attempt integral in $\theta$ only using $x=2\sin\theta$ and $dx = \pm A\cos\theta \, d\theta$ |
| $= \int \frac{1}{4}\sec^2\theta \, d\theta$ | M1 | Uses $1-\sin^2\theta = \cos^2\theta$ and simplifies to $\int C\sec^2\theta \, d\theta$ |
| $= \frac{1}{4}\tan\theta$ | dM1A1 | dM1 dependent on previous M1: $\int \sec^2\theta \to \tan\theta$; A1 for $\frac{1}{4}\tan\theta$ (no $+c$ required) |
| Uses limits $0$ and $\frac{\pi}{3}$ in integrated expression | M1 | Changes $x$-limits to $\theta$-limits of $0$ and $\frac{\pi}{3}$ |
| $= \left[\frac{1}{4}\tan\theta\right]_0^{\pi/3} = \frac{\sqrt{3}}{4}$ | A1 | |
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4. Use the substitution $x = 2 \sin \theta$ to find the exact value of
$$\int _ { 0 } ^ { \sqrt { 3 } } \frac { 1 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$
\hfill \mbox{\textit{Edexcel C34 2015 Q4 [7]}}