Solution 2.2:4d

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m (Lösning 2.2:4d moved to Solution 2.2:4d: Robot: moved page)
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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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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>

Revision as of 12:57, 21 October 2008

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

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