4. $\int \frac{\sin x}{\sqrt{\cos x}} d x$
Substituting back, we have $\boxed{-2 \ln |\sqrt{\cos x}| + C}$
Step 1 :Let $u = \sqrt{\cos x}$, so $u^2 = \cos x$ and $2u \, du = -\sin x \, dx$
Step 2 :Then the integral becomes $-2 \int \frac{1}{u} \, du$
Step 3 :Integrating, we get $-2 \ln |u| + C$
Step 4 :Substituting back, we have $\boxed{-2 \ln |\sqrt{\cos x}| + C}$