Processing Math: Done
To print higher-resolution math symbols, click the
Hi-Res Fonts for Printing button on the jsMath control panel.
No jsMath TeX fonts found -- using image fonts instead.
These may be slow and might not print well.
Use the jsMath control panel to get additional information.
jsMath Control PanelHide this Message

jsMath

Lösung 2.2:3c

Aus Online Mathematik Brückenkurs 2

(Unterschied zwischen Versionen)
Wechseln zu: Navigation, Suche
K (Lösning 2.2:3c moved to Solution 2.2:3c: Robot: moved page)
Zeile 1: Zeile 1:
-
{{NAVCONTENT_START}}
+
It is simpler to investigate the integral if we write it as
-
<center> [[Image:2_2_3c.gif]] </center>
+
 
-
{{NAVCONTENT_STOP}}
+
 
 +
<math>\int{\ln x\centerdot \frac{1}{x}\,dx}</math>,
 +
 
 +
The derivative of
 +
<math>\ln x</math>
 +
is
 +
<math>\frac{1}{x}</math>, so if we choose
 +
<math>u=\ln x</math>, the integral can be expressed as
 +
 
 +
 
 +
<math>\int{u\centerdot {u}'\,dx}</math>
 +
 
 +
 
 +
Thus, it seems that
 +
<math>u=\ln x</math>
 +
is a useful substitution,
 +
 
 +
 
 +
<math>\begin{align}
 +
& \int{\ln x\centerdot \frac{1}{x}\,dx}=\left\{ \begin{matrix}
 +
u=\ln x \\
 +
du=\left( \ln x \right)^{\prime }\,dx=\frac{1}{x}\,dx \\
 +
\end{matrix} \right\} \\
 +
& =\int{u\,du=\frac{1}{2}u^{2}+C} \\
 +
& =\frac{1}{2}\left( \ln x \right)^{2}+C \\
 +
\end{align}</math>

Version vom 11:02, 21. Okt. 2008

It is simpler to investigate the integral if we write it as


lnxx1dx ,

The derivative of lnx is x1, so if we choose u=lnx, the integral can be expressed as


uudx 


Thus, it seems that u=lnx is a useful substitution,


lnxx1dx=u=lnxdu=lnxdx=x1dx=udu=21u2+C=21lnx2+C

Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_2.2:3c“