Question

$\int_{1}^{3} \frac{1}{x(1+x^2)} dx$

Answer 1

$\ln\left(\frac{3}{\sqrt{5}}\right)$

Answer 2

The integral $\int_{1}^{3} \frac{1}{x(1+x^2)} dx$ is evaluated using partial fraction decomposition. The integrand is split into $\frac{1}{x} - \frac{x}{1+x^2}$. Each term is then integrated: $\int \frac{1}{x} dx = \ln|x|$ and $\int \frac{x}{1+x^2} dx = \frac{1}{2} \ln(1+x^2)$ (using substitution $u=1+x^2$). The definite integral is then evaluated by substituting the limits of integration and simplifying the logarithmic expression using properties of logarithms.