4. $\int \frac{\sin x}{\sqrt{\cos x}} d x$

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 $\boxed{-2 \ln |\sqrt{\cos x}| + C}$