To evaluate the integral \( \int \frac{1}{1 + \sin x} \, dx \), we will follow these steps:
### Step 1: Rationalize the Denominator
We start by multiplying the numerator and the denominator by \( 1 - \sin x \):
\[
\int \frac{1}{1 + \sin x} \, dx = \int \frac{1 - \sin x}{(1 + \sin x)(1 - \sin x)} \, dx
\]
This simplifies to:
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