Determine the value of `int_(-pi)^(pi) (2x(1+sinx))/(1+cos^(2)x)dx`.

Explanation

To solve the integral \( I = \int_{-\pi}^{\pi} \frac{2x(1+\sin x)}{1+\cos^2 x} \, dx \), we can break it down into two parts based on the properties of even and odd functions.

### Step-by-Step Solution:

1. **Define the Integral**:
\[ I = \int_{-\pi}^{\pi} \frac{2x(1+\sin x)}{1+\cos^2 x} \, dx \]
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