Let's delve into how to derive the integral: $\int_{0}^{1} \frac{\arctan(x)}{1+x} \, dx$

This integral is not straightforward, and while it can be evaluated using advanced techniques like series expansions or recognizing it as a special function integral, I'll outline a method that gives us the result based on known integrals.

Method: Recognizing a Known Integral

1. Rewrite the Integral: We aim to evaluate: $I = \int_{0}^{1} \frac{\arctan(x)}{1+x} \, dx$ This integral doesn't simplify easily using elementary methods like substitution or integration by parts, but it's known in mathematical tables that: $\int_{0}^{1} \frac{\arctan(x)}{1+x} \, dx = \frac{\pi}{8} \ln(2)$

2. Understanding the Result: The result $\frac{\pi}{8} \ln(2)$ comes from deeper analysis involving the properties of the arctangent function and the logarithmic function.

   To briefly outline why this is true:

   • The arctangent function ( \arctan(x)