Question

∫1/x(x^2+1)

Answer 1

ln|x| - 1/2ln(x^2+1) + C

Answer 2

The integral $\int \frac{1}{x(x^2+1)} dx$ is solved by partial fraction decomposition. The integrand is split into $\frac{1}{x} - \frac{x}{x^2+1}$. Integrating $\frac{1}{x}$ gives $\ln|x|$. Integrating $\frac{x}{x^2+1}$ uses a substitution $u=x^2+1$, leading to $\frac{1}{2}\ln(x^2+1)$. Combining these results yields $\ln|x| - \frac{1}{2}\ln(x^2+1) + C$.