Find ∫ 1/(x(1+x^2)) dx.

The correct answer is Option (3) → $\ln|x| - \frac{1}{2}(1+x^2) + C$

$I = \int \frac{1}{x(1+x^2)} dx = \int \left( \frac{1}{x} - \frac{x}{1+x^2} \right) dx$

$= \log |x| - \frac{1}{2} \log(1+x^2) + C$