Example 3.5.1.

Evaluate \(\hspace {0.4cm} \displaystyle {I = \int \dfrac {x^4 - x^2 - 1}{x^2 (x - 1)}\hspace {0.1cm} dx}\)

\begin {align*} I & = \int \Bigg ( x + \frac {2}{x} + \frac {1}{x^2} - \frac {2}{x - 1}\Bigg )\hspace {0.1cm} dx\\\\ & = \int x\hspace {0.1cm}dx + \int \frac {2}{x}\hspace {0.1cm}dx + \int \frac {1}{x^2}\hspace {0.1cm}dx - \int \frac {2}{x - 1}\hspace {0.1cm}dx\\\\ & = \int x\hspace {0.1cm}dx + 2\int \frac {1}{x}\hspace {0.1cm}dx + \int \frac {1}{x^2}\hspace {0.1cm}dx -2 \int \frac {1}{x - 1}\hspace {0.1cm}dx\\\\ & = \dfrac {1}{2}x^2 + 2\ln |x| - \dfrac {1}{x} - 2\ln |x - 1| + c\\\\ \end {align*}