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