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Lösung 2.2:4d

Aus Online Mathematik Brückenkurs 2

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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 <math>x^2</math>-term from the numerator
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 <math>x^2</math>-term from the numerator
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{{Displayed math||<math>\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{.}</math>}}
+
{{Abgesetzte Formel||<math>\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{.}</math>}}
Thus, we have
Thus, we have
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{{Displayed math||<math>\int\frac{x^2}{x^2+1}\,dx = \int\Bigl(1-\frac{1}{x^2+1} \Bigr)\,dx = x-\arctan x+C\,\textrm{.}</math>}}
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{{Abgesetzte Formel||<math>\int\frac{x^2}{x^2+1}\,dx = \int\Bigl(1-\frac{1}{x^2+1} \Bigr)\,dx = x-\arctan x+C\,\textrm{.}</math>}}

Version vom 13:03, 10. 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 x2-term from the numerator

x2x2+1=x2+1x2+1−1=x2+1x2+1−1x2+1=1−1x2+1.

Thus, we have

x2x2+1dx=1−1x2+1dx=x−arctanx+C. 
Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_2.2:4d“