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Lösung 2.1:2c

Aus Online Mathematik Brückenkurs 2

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K (Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel))
Zeile 1: Zeile 1:
If we recall that <math>\sqrt{x} = x^{1/2}</math>, the integral can be written as
If we recall that <math>\sqrt{x} = x^{1/2}</math>, the integral can be written as
-
{{Displayed math||<math>\begin{align}
+
{{Abgesetzte Formel||<math>\begin{align}
\int\limits_{4}^{9} \bigl(\sqrt{x}-\frac{1}{\sqrt{x}}\Bigr)\,dx
\int\limits_{4}^{9} \bigl(\sqrt{x}-\frac{1}{\sqrt{x}}\Bigr)\,dx
&= \int\limits_{4}^{9}\Bigl( x^{1/2}-\frac{1}{x^{1/2}}\Bigr)\,dx\\[5pt]
&= \int\limits_{4}^{9}\Bigl( x^{1/2}-\frac{1}{x^{1/2}}\Bigr)\,dx\\[5pt]
Zeile 11: Zeile 11:
We obtain
We obtain
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{{Displayed math||<math>\begin{align}
+
{{Abgesetzte Formel||<math>\begin{align}
\int\limits_{4}^{9} \bigl( x^{1/2}-x^{-1/2}\bigr)\,dx
\int\limits_{4}^{9} \bigl( x^{1/2}-x^{-1/2}\bigr)\,dx
&= \Bigl[\ \frac{x^{1/2+1}}{1/2+1} - \frac{x^{-1/2+1}}{-1/2+1}\ \Bigr]_{4}^{9}\\[5pt]
&= \Bigl[\ \frac{x^{1/2+1}}{1/2+1} - \frac{x^{-1/2+1}}{-1/2+1}\ \Bigr]_{4}^{9}\\[5pt]

Version vom 12:58, 10. Mär. 2009

If we recall that x=x12 , the integral can be written as

94x−1xdx=94x12−1x12dx=94x12−x−12dx.

This is a standard integral in which the integrand consists of two terms looking like xn, where n=12 and n=−12, respectively.

We obtain

94x12−x−12dx= x12+112+1−x−12+1−12+1 94= 32x1+12−12x12 94= 32xx−2x 94=3299−29−3244−24=3293−23−3242−22=18−6−316+4=16−316=3163−16=332.
Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_2.1:2c“