Function Expression :
\[f(x)=\frac{ln x}{x+2} \]Domain
\[\left]0, \infty\right[ \]Limits
\[\lim_{x \overset{>}{\rightarrow0} }f(x) = -\infty \]\[\lim_{x \rightarrow+\infty}f(x) = 0 \]
\[ \]
Derivate
\[f^{\,\prime}(x)=- \frac{\log{\left(x \right)}}{\left(x + 2\right)^{2}} + \frac{1}{x \left(x + 2\right)} \]\[f^{\,\prime}(x)=\frac{- x \log{\left(x \right)} + x + 2}{x \left(x + 2\right)^{2}} \]
\[ \]
Integral
\[F(x) = \begin{cases} - \operatorname{Li}_{2}\left(\frac{x}{2} + 1\right) & \text{for}\: \frac{1}{\left|{x + 2}\right|} < 1 \wedge \left|{x + 2}\right| < 1 \\\log{\left(2 \right)} \log{\left(x + 2 \right)} + 3 i \pi \log{\left(x + 2 \right)} - \operatorname{Li}_{2}\left(\frac{x}{2} + 1\right) & \text{for}\: \left|{x + 2}\right| < 1 \\- \log{\left(2 \right)} \log{\left(\frac{1}{x + 2} \right)} - 3 i \pi \log{\left(\frac{1}{x + 2} \right)} - \operatorname{Li}_{2}\left(\frac{x}{2} + 1\right) & \text{for}\: \frac{1}{\left|{x + 2}\right|} < 1 \\- {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x + 2} \right)} \log{\left(2 \right)} - 3 i \pi {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x + 2} \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x + 2} \right)} \log{\left(2 \right)} + 3 i \pi {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x + 2} \right)} - \operatorname{Li}_{2}\left(\frac{x}{2} + 1\right) & \text{otherwise} \end{cases} \]Sign Table
Variation Table
Plot
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