Function Expression :
\[f(x)=x-e+ln(1+2e^{-2(x-e )} ) \]Domain
\[\left]-\infty, \infty\right[ \]Limits
\[\lim_{x \rightarrow-\infty}f(x) = +\infty \]\[\lim_{x \rightarrow+\infty}f(x) = +\infty \]
\[ \]
Derivate
\[f^{\,\prime}(x)=1 - \frac{4 e^{- 2 x + 2 e}}{1 + 2 e^{\left(-1\right) 2 \left(x - e\right)}} \]\[f^{\,\prime}(x)=\frac{e^{2 x} - 2 e^{2 e}}{e^{2 x} + 2 e^{2 e}} \]
\[ \]
Integral
\[F(x) = x \log{\left(2 e^{- 2 x + 2 e} + 1 \right)} + \int \frac{x e^{2 x} + 6 x e^{2 e} - e e^{2 x} - 2 e e^{2 e}}{e^{2 x} + 2 e^{2 e}}\, dx \]Sign Table
Variation Table
Plot
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