Lösung 2.2:4a

Aus Online Mathematik Brückenkurs 2

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K (Lösning 2.2:4a moved to Solution 2.2:4a: Robot: moved page)
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What makes our integral differ from that in the exercise´s text is that there is a term
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<center> [[Image:2_2_4a.gif]] </center>
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<math>x^{2}+4</math>
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instead of
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<math>x^{2}+1</math>, but if we factor out the 4 from the denominator,
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<math>\int{\frac{\,dx}{x^{2}+4}}=\int{\frac{\,dx}{4\left( \frac{1}{4}x^{2}+1 \right)}}=\frac{1}{4}\int{\frac{\,dx}{\frac{1}{4}x^{2}+1}}</math>,
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we obtain the correct second term in the denominator. On the other hand, there is no longer
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<math>x^{2}</math>
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but
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<math>\frac{1}{4}x^{2}</math>, although we can get around this by substituting
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<math>u=\frac{1}{2}x</math>,
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<math>\begin{align}
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& \frac{1}{4}\int{\frac{\,dx}{\frac{1}{4}x^{2}+1}}=\frac{1}{4}\int{\frac{\,dx}{\left( \frac{x}{2} \right)^{2}+1}}=\left\{ \begin{matrix}
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u=\frac{1}{2}x \\
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du=\frac{1}{2}\,dx \\
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\end{matrix} \right\} \\
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& =\frac{1}{4}\int{\frac{2\,du}{u^{2}+1}}=\frac{1}{2}\int{\frac{\,du}{u^{2}+1}} \\
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& =\frac{1}{2}\arctan u+C=\frac{1}{2}\arctan \frac{x}{2}+C \\
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\end{align}</math>

Version vom 12:22, 21. Okt. 2008

What makes our integral differ from that in the exercise´s text is that there is a term \displaystyle x^{2}+4 instead of \displaystyle x^{2}+1, but if we factor out the 4 from the denominator,


\displaystyle \int{\frac{\,dx}{x^{2}+4}}=\int{\frac{\,dx}{4\left( \frac{1}{4}x^{2}+1 \right)}}=\frac{1}{4}\int{\frac{\,dx}{\frac{1}{4}x^{2}+1}},

we obtain the correct second term in the denominator. On the other hand, there is no longer \displaystyle x^{2} but \displaystyle \frac{1}{4}x^{2}, although we can get around this by substituting \displaystyle u=\frac{1}{2}x,


\displaystyle \begin{align} & \frac{1}{4}\int{\frac{\,dx}{\frac{1}{4}x^{2}+1}}=\frac{1}{4}\int{\frac{\,dx}{\left( \frac{x}{2} \right)^{2}+1}}=\left\{ \begin{matrix} u=\frac{1}{2}x \\ du=\frac{1}{2}\,dx \\ \end{matrix} \right\} \\ & =\frac{1}{4}\int{\frac{2\,du}{u^{2}+1}}=\frac{1}{2}\int{\frac{\,du}{u^{2}+1}} \\ & =\frac{1}{2}\arctan u+C=\frac{1}{2}\arctan \frac{x}{2}+C \\ \end{align}

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_2.2:4a“