<p>Evaluate \(\displaystyle\int_0^1\frac{dx}{\sqrt{x(1-x)}}\)</p>
Step-by-Step Solution
Key Concept: Complete the square: x(1-x) = 1/4 - (x-1/2)^2. Then sub x-1/2 = (1/2)sin \theta.
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<p>$x(1-x) = \frac{1}{4}-(x-\frac{1}{2})^2$. Let $x-\frac{1}{2}=\frac{1}{2}\sin\theta$, $dx=\frac{1}{2}\cos\theta\,d\theta$.</p>
<p>Limits: $x=0\Rightarrow\theta=-\pi/2$; $x=1\Rightarrow\theta=\pi/2$.</p>
<p>$$\int_{-\pi/2}^{\pi/2}\frac{\frac{1}{2}\cos\theta\,d\theta}{\sqrt{\frac{1}{4}\cos^2\theta}} = \int_{-\pi/2}^{\pi/2}\frac{\frac{1}{2}\cos\theta}{\frac{1}{2}\cos\theta}\,d\theta = \int_{-\pi/2}^{\pi/2}d\theta = \boxed{\pi}$$</p>
<p><em>Alternatively:</em> Beta function $B(1/2,1/2) = \Gamma(1/2)^2/\Gamma(1) = \pi$.</p>
Correct Answer: B