Function Expression :
\[f(x)=x+\frac{1}{2}+2ln(\frac{x}{x-1} ) \]Domain
\[\left]-\infty, 0\right[ \cup \left]1, \infty\right[ \]Limits
\[\lim_{x \rightarrow-\infty}f(x) = -\infty \]\[\lim_{x \overset{<}{\rightarrow0} }f(x) = -\infty \]
\[\lim_{x \overset{>}{\rightarrow1} }f(x) = +\infty \]
\[\lim_{x \rightarrow+\infty}f(x) = +\infty \]
\[ \]
Derivate
\[f^{\,\prime}(x)=1 + \frac{2 \left(x - 1\right) \left(- \frac{x}{\left(x - 1\right)^{2}} + \frac{1}{x - 1}\right)}{x} \]\[f^{\,\prime}(x)=\frac{x \left(x - 1\right) - 2}{x \left(x - 1\right)} \]
\[ \]
Integral
\[F(x) = \frac{x^{2}}{2} + 2 x \log{\left(\frac{x}{x - 1} \right)} + \frac{x}{2} + 2 \log{\left(x - 1 \right)} \]Sign Table
Variation Table
Plot
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