Answer
$$
\int \frac{\cos x}{\sqrt {1+\sin x}} d x= 2 (1+\sin x)^{\frac{1}{2}}+C
$$
Work Step by Step
$$
\int \frac{\cos x}{\sqrt {1+\sin x}} d x
$$
Let $ u=1+\sin x $. Then $ du= \cos x dx $, so
$$
\begin{split}
\int \frac{\cos x}{\sqrt {1+\sin x}} d x & =\int u^{-\frac{1}{2}}d u
\\
& =2 u^{\frac{1}{2}}+C
\\
& =2 (1+\sin x)^{\frac{1}{2}}+C.
\end{split}
$$
