Question: $\frac{\cos x}{\sqrt{1+\sin x}}$
Solution:
Let $1+\sin x=t$
$\therefore \cos x d x=d t$
$\Rightarrow \int \frac{\cos x}{\sqrt{1+\sin x}} d x=\int \frac{d t}{\sqrt{t}}$
$=\frac{t^{\frac{1}{2}}}{\frac{1}{2}}+\mathrm{C}$
$=2 \sqrt{t}+\mathrm{C}$
$=2 \sqrt{1+\sin x}+\mathrm{C}$
