Evaluate: `int1/(1+2cosx)\ dx`

Explanation

To evaluate the integral \( I = \int \frac{1}{1 + 2 \cos x} \, dx \), we can follow these steps:

### Step 1: Rewrite the Cosine Function
We start by rewriting \( \cos x \) using the half-angle identity. The half-angle identity states that:
\[ \cos x = \frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)} \]
Thus, we can express \( 2 \cos x \) as:
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