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Lösung 2.2:4d

Aus Online Mathematik Brückenkurs 2

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K (Lösning 2.2:4d moved to Solution 2.2:4d: Robot: moved page)
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{{NAVCONTENT_START}}
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The integral can be simplified by a so-called polynomial division. We add and take away
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<center> [[Image:2_2_4d.gif]] </center>
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<math>\text{1}</math>
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{{NAVCONTENT_STOP}}
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in the numerator and can thus eliminate the
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<math>x^{2}</math>
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-term from the numerator
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<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}</math>
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Thus, we have
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<math>\int{\frac{x^{2}}{x^{2}+1}\,dx=\int{\left( 1-\frac{1}{x^{2}+1} \right)}}\,dx=x-\arctan x+C</math>

Version vom 12:57, 21. Okt. 2008

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“