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Question
\[\int\limits_0^{\pi/2} \frac{\sin x}{\sqrt{1 + \cos x}} dx\]
Solution
\[\int_0^\frac{\pi}{2} \frac{\sin x}{\sqrt{1 + \cos x}} d x\]
\[Let 1 + \cos x = t, then - \sin x dx = dt\]
\[When, x \to 0, t \to 2 and x \to \frac{\pi}{2}, t \to 1\]
Therefore, the integral becomes
\[ \int_2^1 \frac{- 1}{\sqrt{t}}dt\]
\[ = \int_1^2 \frac{1}{\sqrt{t}}dt\]
\[ = 2 \left[ \sqrt{t} \right]_1^2 \]
\[ = 2\left( \sqrt{2} - 1 \right)\]
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