Aus Online Mathematik Brückenkurs 2

By dividing the two terms in the numerator by \displaystyle x, we can simplify each term to a form which makes it possible simply to write down the primitive functions of the integrand:

\displaystyle \begin{align} & \int{\frac{x^{2}+1}{x}}\,dx=\int{\left( \frac{x^{2}}{x}+\frac{1}{x} \right)}\,dx \\ & =\int{\left( x+x^{-1} \right)}\,dx \\ & =\frac{x^{2}}{2}+\ln \left| x \right|+C \\ \end{align}

where \displaystyle C is an arbitrary constant.

NOTE: observe that \displaystyle \frac{1}{x} has a singularity at \displaystyle x=0, so the answers above are only primitive functions over intervals that do not contain \displaystyle x=0.