Question

\[\int\limits_0^{\pi/2} \frac{\sin \theta}{\sqrt{1 + \cos \theta}} d\theta\]

Answer 1

\[Let\ I = \int_0^\frac{\pi}{2} \frac{\sin \theta}{\sqrt{1 + \cos \theta}} d \theta . \]

\[Let\ \cos\ \theta = t . Then, - \sin\ \theta\ d\theta\ = dt\]

\[When\ \theta = 0, t = 1\ and\ \theta = \frac{\pi}{2}, t = 0\]

\[ \therefore I = \int_0^\frac{\pi}{2} \frac{\sin \theta}{\sqrt{1 + \cos \theta}} d \theta\]

\[ = \int_1^0 \frac{- dt}{\sqrt{1 + t}}\]

\[ = \int_0^1 \frac{dt}{\sqrt{1 + t}}\]

\[ = 2 \left[ \sqrt{1 + t} \right]_0^1 \]

\[ = 2\left( \sqrt{2} - 1 \right)\]