Answer
$$\frac{2 \sqrt{\sin x+1}}{3}(\sin x-2)+C$$
Work Step by Step
Given $$\int \frac{\sin x \cdot \cos x}{\sqrt{\sin x+1}} d x$$
Let
$$ u=\sin x+1\ \ \ \ \Rightarrow \ \ \ du = \cos xdx$$
Then
\begin{aligned} \int \frac{\sin x \cdot \cos x}{\sqrt{\sin x+1}} d x &=\int \frac{u-1}{\sqrt{u}} d u \\ &=\int\left(u^{1 / 2}-u^{-1 / 2}\right) d u \\ &=\frac{2 u^{3 / 2}}{3}-2 \cdot u^{1 / 2}+C \\ &=\frac{\sqrt{\sin x+1}}{3}(2 \sin x-4)+C \\ &=\frac{2 \sqrt{\sin x+1}}{3}(\sin x-2)+C \end{aligned}
