Evaluate : `int(dx)/(sqrt(1-sinx))`

Explanation

To evaluate the integral \( I = \int \frac{dx}{\sqrt{1 - \sin x}} \), we will follow a series of steps involving trigonometric identities and substitutions.

### Step 1: Rewrite the Denominator
We start by rewriting the denominator using the identity \( 1 - \sin x = 1 - \frac{2 \sin \frac{x}{2} \cos \frac{x}{2}}{1} \). Thus, we can express \( 1 - \sin x \) in terms of \( \sin \frac{x}{2} \) and \( \cos \frac{x}{2} \):
\[ 1 - \sin x = 1 - 2 \sin \frac{x}{2} \cos \frac{x}{2} = \left(\sin \frac{x}{2} - \cos \frac{x}{2}\right)^2 \]

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