Aus Online Mathematik Brückenkurs 2

The integral can be simplified by a so-called polynomial division. We add and take away \displaystyle \text{1} in the numerator and can thus eliminate the \displaystyle x^{2} -term from the numerator

\displaystyle \frac{x^{2}}{x^{2}+1}=\frac{x^{2}+1-1}{x^{2}+1}=\frac{x^{2}+1}{x^{2}+1}-\frac{1}{x^{2}+1}=1-\frac{1}{x^{2}+1}

Thus, we have

\displaystyle \int{\frac{x^{2}}{x^{2}+1}\,dx=\int{\left( 1-\frac{1}{x^{2}+1} \right)}}\,dx=x-\arctan x+C