Function Expression :
\[f(x)=\frac{e^x}{e^x-1}ln(e^x-1 ) \]Domain
\[\left]0, \infty\right[ \]Limits
\[\lim_{x \overset{>}{\rightarrow0} }f(x) = -\infty \]\[\lim_{x \rightarrow+\infty}f(x) = +\infty \]
\[ \]
Derivate
\[f^{\,\prime}(x)=\frac{e^{x} \log{\left(e^{x} - 1 \right)}}{e^{x} - 1} - \frac{e^{2 x} \log{\left(e^{x} - 1 \right)}}{\left(e^{x} - 1\right)^{2}} + \frac{e^{2 x}}{\left(e^{x} - 1\right)^{2}} \]\[f^{\,\prime}(x)=\frac{\left(e^{x} - \log{\left(e^{x} - 1 \right)}\right) e^{x}}{e^{2 x} - 2 e^{x} + 1} \]
\[ \]
Integral
\[F(x) = \frac{\log{\left(e^{x} - 1 \right)}^{2}}{2} \]Sign Table
Variation Table
Plot
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