\(\rm \int \frac{dx}{x(x^2 + 1)}\) किसके बराबर है?

प्रयुक्त सूत्र:

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

logax - logay = \(\rm log_{a}\frac{x}{y}\)

गणना:

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

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

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

= 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

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

∴\(\int \frac{dx}{x(x^2 + 1)}\)  

\(\rm \frac{1}{2}ln\left(\frac{x^2}{x^2 + 1}\right) + C\) के बराबर है|