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