Lösung 2.2:4d
Aus Online Mathematik Brückenkurs 2
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Version vom 10:25, 11. Mär. 2009
The integral can be simplified by a so-called polynomial division. We add and take away 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}\,\textrm{.} |
Thus, we have
| \displaystyle \int\frac{x^2}{x^2+1}\,dx = \int\Bigl(1-\frac{1}{x^2+1} \Bigr)\,dx = x-\arctan x+C\,\textrm{.} |
