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 \Bigl(\frac{x^2}{x} + \frac{1}{x}\Bigr)\,dx\\[5pt] &= \int \bigl(x+x^{-1}\bigr)\,dx\\[5pt] &= \frac{x^2}{2} + \ln |x| + C\,, \end{align}

where \displaystyle C is an arbitrary constant.

Note: Observe that \displaystyle 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\,.