Let $u = \cos x$. Then, $du = -\sin x \, dx$.
We can rewrite the integral as:
$$\int \frac{\sin x}{\sqrt{\cos x}} \, dx = \int \frac{-du}{\sqrt{u}}$$
Now, we can simplify the integral:
$$\int \frac{-du}{\sqrt{u}} = -\int u^{-\frac{1}{2}} \, du$$
Using
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