Standard +0.3 This is a straightforward application of trigonometric substitution with clear guidance (the substitution is given). Students must find dx/dθ, substitute to get sec²θ, change limits (0 to π/6), and integrate to get tan θ. While it requires careful algebraic manipulation and exact value recall (sin π/6 = 1/2, tan π/6 = 1/√3), it's a standard C4 technique with no novel problem-solving required, making it slightly easier than average.
$\int\frac{1}{(1-x^2)^{\frac{1}{2}}}dx = \int\frac{1}{(1-\sin^2\theta)^{\frac{1}{2}}}\cos\theta\,d\theta$ | M1 |
Use of $x = \sin\theta$ and $\frac{dx}{d\theta} = \cos\theta$ | |
$= \int\frac{1}{\cos^2\theta}d\theta$ | M1 A1 |
$= \int\sec^2\theta\,d\theta = \tan\theta$ | M1 A1 |
Using the limits $0$ and $\frac{\pi}{6}$ to evaluate integral | M1 |
$[\tan\theta]_0^{\pi/6} = \frac{1}{\sqrt{3}} \left(= \frac{\sqrt{3}}{3}\right)$ | cao A1 |
**Alternative for final M1 A1:**
Returning to the variable $x$ and using the limits $0$ and $\frac{1}{2}$ to evaluate integral | M1 |
$\left[\frac{x}{\sqrt{1-x^2}}\right]_0^{1/2} = \frac{1}{\sqrt{3}} \left(= \frac{\sqrt{3}}{3}\right)$ | cao A1 |
**Total: [7]**
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