What is \(\rm \int \frac{dx}{x(x^2 + 1)}\) equal to?

Formula used:

\(\rm \int \frac{1}{x} dx = log \: x + C\)

log_ax - log_ay = \(\rm log_{a}\frac{x}{y}\)

Calculation:

\(\int \frac{dx}{x(x^2 + 1)}\)

⇒ \(\rm \int \frac{dx}{x} - \frac{xdx}{x^{2} + 1}\)

⇒ \(\rm \int \frac{dx}{x} - \frac{1}{2}\int \frac{2xdx}{x^{2} + 1}\)

⇒ ln x - \(\rm \frac{1}{2} ln |1 + x^{2}|\) + c

⇒ \(\frac{1}{2} \ln x^{2} -\)\(\rm \frac{1}{2} ln |1 + x^{2}|\) + c

⇒ \(\rm \frac{1}{2}ln\left(\frac{x^2}{x^2 + 1}\right) + C\)

∴ \(\int \frac{dx}{x(x^2 + 1)}\) is equal to 

\(\rm \frac{1}{2}ln\left(\frac{x^2}{x^2 + 1}\right) + C\).