Function Expression :
\[f(x)=x-1-\frac{ln(x+1 )}{x+1} \]Domain
\[\left]-1, \infty\right[ \]Limits
\[\lim_{x \overset{>}{\rightarrow-1} }f(x) = +\infty \]\[\lim_{x \rightarrow+\infty}f(x) = +\infty \]
\[ \]
Derivate
\[f^{\,\prime}(x)=1 + \frac{\log{\left(x + 1 \right)}}{\left(x + 1\right)^{2}} - \frac{1}{\left(x + 1\right)^{2}} \]\[f^{\,\prime}(x)=\frac{\left(x + 1\right)^{2} + \log{\left(x + 1 \right)} - 1}{\left(x + 1\right)^{2}} \]
\[ \]
Integral
\[F(x) = \frac{x^{2}}{2} - x - \begin{cases} \frac{\log{\left(x + 1 \right)}^{2}}{2} & \text{for}\: \left|{x + 1}\right| < 1 \\\frac{\log{\left(\frac{1}{x + 1} \right)}^{2}}{2} & \text{for}\: \frac{1}{\left|{x + 1}\right|} < 1 \\{G_{3, 3}^{3, 0}\left(\begin{matrix} & 1, 1, 1 \\0, 0, 0 & \end{matrix} \middle| {x + 1} \right)} + {G_{3, 3}^{0, 3}\left(\begin{matrix} 1, 1, 1 & \\ & 0, 0, 0 \end{matrix} \middle| {x + 1} \right)} & \text{otherwise} \end{cases} \]Sign Table
Variation Table
Plot
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