Evaluate: `int_0^pilog(1+cosx)dx`

Explanation

To evaluate the integral \( I = \int_0^\pi \log(1 + \cos x) \, dx \), we will use properties of definite integrals and logarithmic identities. Here’s a step-by-step solution:

### Step 1: Use the property of definite integrals
We know that:
\[ \int_0^a f(x) \, dx = \int_0^a f(a - x) \, dx \]
For our case, we will apply this property with \( a = \pi \):
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