Question

Find \(\int_{-1}^{1} \frac{\log(1 + |x|)}{1 + |x|} \, dx\)

Hint:

When dealing with absolute values in integrals, split the integral based on the behavior of the function on the different parts of the domain.

Answer 1

Step 1: Break the integral into two parts.
Since the integrand involves \(|x|\), we split the integral at 0: \[ \int_{-1}^{1} \frac{\log(1 + |x|)}{1 + |x|} \, dx = \int_{-1}^{0} \frac{\log(1 - x)}{1 - x} \, dx + \int_{0}^{1} \frac{\log(1 + x)}{1 + x} \, dx \]
Step 2: Use symmetry.
The integrals on the intervals \([-1, 0]\) and \([0, 1]\) are symmetric because of the absolute value in the function. Thus, both parts contribute equally to the value of the integral.
Step 3: Solve the integral.
Both integrals can be solved using standard integration techniques or known integral results. After performing the integration, the result is: \[ \boxed{0} \]