`intcos^2x/2\ dx`

Explanation

To solve the integral \( \int \cos^2\left(\frac{x}{2}\right) \, dx \), we can use the half-angle identity for cosine. Here's the step-by-step solution:

### Step 1: Use the Half-Angle Formula
The half-angle formula states that:
\[ \cos^2\left(\frac{x}{2}\right) = \frac{1 + \cos(x)}{2} \]
Thus, we can rewrite the integral as:
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