Differentialrechnung

(Difference between revisions)

Revision as of 07:46, 17 September 2008

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 |<math>\frac 1{\sqrt{x^2-1}}</math>
-|<math>\operatorname{arcosh} x+C = \ln\, \left|x+\sqrt{x^2-1}\right|+C</math>
+|<math>\operatorname{arcosh} x+C = \ln\, \left|x+\sqrt{x^2-1}\right|+C,\ |x|>1</math>
 |-
 |<math>\frac 1{\sqrt{1-x^2}}</math>
-|<math>\operatorname{artanh} x+C = \frac 1 2 \ln\frac{1+x}{1-x}+C</math>
+|<math>\begin{cases}\operatorname{artanh}x+C=\frac 1 2 \ln\frac{1+x}{1-x}+C,& |x|<1\\\operatorname{arcoth}x+C=\frac 1 2 \ln\frac{x+1}{x-1}+C,& |x|>1\end{cases}</math>
-|-
-|<math>\frac 1{\sqrt{1-x^2}}</math>
-|<math>\operatorname{arcoth} x+C = \frac 1 2 \ln\frac{x+1}{x-1}+C</math>
 |-
 |<math></math>