Evaluate : `int_0^1xlog(1+2x)dx`

Explanation

To evaluate the integral \( I = \int_0^1 x \log(1 + 2x) \, dx \), we can use integration by parts.

### Step 1: Identify the parts for integration by parts
Let:
- \( u = \log(1 + 2x) \) (first function)
- \( dv = x \, dx \) (second function)

### Step 2: Differentiate and integrate
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