Question

$\int \frac{\cos x}{\sqrt{1+\sin x}}dx$

Answer 1

We have,

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

Let $t=1+\sin x$

$dt=\cos xdx$

Therefore,

$I=\int \frac{1}{\sqrt{t}}dt$

$I=2\sqrt{t}+C$

On putting the value of $t$, we get

$I=2\sqrt{1+\sin x}+C$

Hence, this is the answer.