Question

Show that $\int \frac{\sin x}{\sqrt{(1+\sin x)}}dx=\int (\cos \frac{x}{2}+\sin \frac{x}{2})dx+\int \frac{dx}{\cos \left(x/2\right)+\sin \left(x/2\right)}.$

Answer 1

$I=\int \frac{\sin x}{\sqrt{(1+\sin x)}}dx$

$=\int \frac{(1+\sin x)-1}{\sqrt{(1+\sin x)}}dx$

$=\int \sqrt{1+\sin x}dx-\int \frac{1}{\sqrt{(1+\sin x)}}dx$

$=\int \sqrt{{(\cos \frac{x}{2}+\sin \frac{x}{2})}^2}dx-\int \frac{dx}{\sqrt{{(\cos \frac{x}{2}+\sin \frac{x}{2})}^2}}$

$=\int (\cos \frac{x}{2}+\sin \frac{x}{2})dx-\int \frac{dx}{\cos \frac{x}{2}+\sin \frac{x}{2}}$