Question

Show that $\int \frac{\sin \alpha }{\sqrt{(1+\cos \alpha )}}d\alpha =-2\sqrt{2}\cos \frac{\alpha }{2}$

Answer 1

Let $I=\int \frac{\sin \alpha }{\sqrt{(1+\cos \alpha )}}d\alpha$

Put $\cos \alpha =t$

$\Rightarrow -\sin \alpha d\alpha =dt$

$\therefore I=\int \frac{-1}{\sqrt{1+t}}dt$

$=-2\sqrt{1+t}+C$

$=-2\sqrt{1+\cos \alpha }+C$

$=-2\sqrt{1+2{\cos }^2\frac{\alpha }{2}-1}+C$

$\Rightarrow =-2\sqrt{2}\cos \frac{\alpha }{2}+C$