Question

Evaluate $\int \frac{\sin xdx}{\sqrt{(1+\sin x})}$:

Answer 1

$\begin{array}{c}{\int }^{}\frac{\sin x}{1+\sin x}={\int }^{}\left(\frac{\sin \left(x\right)+1}{\sin \left(x\right)+1}-\frac{1}{\sin \left(x\right)+1}\right)dx\\ ={\int }^{}\left(1-\frac{1}{\sin \left(x\right)+1}\right)dx\\ ={\int }^{}11mudx{-}^{\frac{1}{\sin \left(x\right)+1}}dx1mu\\ =x+\frac{2}{\tan \left(\frac{x}{2}\right)+1}+C\end{array}$

Hence, the integral is $x+\frac{2}{\tan \left(\frac{x}{2}\right)+1}+C$