Function Expression :
\[f(x)=ln(\sqrt{x-2}-3 ) \]Domain
\[\left]11, \infty\right[ \]Limits
\[\lim_{x \overset{>}{\rightarrow11} }f(x) = -\infty \]\[\lim_{x \rightarrow+\infty}f(x) = +\infty \]
\[ \]
Derivate
\[f^{\,\prime}(x)=\frac{1}{2 \sqrt{x - 2} \left(\sqrt{x - 2} - 3\right)} \]\[f^{\,\prime}(x)=\frac{1}{2 \left(x - 3 \sqrt{x - 2} - 2\right)} \]
\[ \]
Integral
\[F(x) = \begin{cases} - \frac{x}{2} - 3 \sqrt{x - 2} + \left(x - 2\right) \log{\left(\frac{\sqrt{x - 2}}{3} - 1 \right)} + \left(x - 2\right) \log{\left(3 \right)} + 2 i \pi \left(x - 2\right) - 9 \log{\left(\frac{\sqrt{x - 2}}{3} - 1 \right)} + 1 & \text{for}\: \left|{\sqrt{x - 2}}\right| > 3 \wedge \left|{x - 2}\right| < 1 \\- \frac{x}{2} - 3 \sqrt{x - 2} + \left(x - 2\right) \log{\left(1 - \frac{\sqrt{x - 2}}{3} \right)} + \left(x - 2\right) \log{\left(3 \right)} + 3 i \pi \left(x - 2\right) - 9 \log{\left(1 - \frac{\sqrt{x - 2}}{3} \right)} + 1 & \text{for}\: \left|{x - 2}\right| < 1 \\- \frac{x}{2} - 3 \sqrt{x - 2} + \left(x - 2\right) \log{\left(\frac{\sqrt{x - 2}}{3} - 1 \right)} - i \pi \left(x - 2\right) + {G_{2, 2}^{1, 1}\left(\begin{matrix} 1 & 2 \\1 & 0 \end{matrix} \middle| {x - 2} \right)} \log{\left(3 \right)} + 3 i \pi {G_{2, 2}^{1, 1}\left(\begin{matrix} 1 & 2 \\1 & 0 \end{matrix} \middle| {x - 2} \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} 2, 1 & \\ & 1, 0 \end{matrix} \middle| {x - 2} \right)} \log{\left(3 \right)} + 3 i \pi {G_{2, 2}^{0, 2}\left(\begin{matrix} 2, 1 & \\ & 1, 0 \end{matrix} \middle| {x - 2} \right)} - 9 \log{\left(\frac{\sqrt{x - 2}}{3} - 1 \right)} + 1 & \text{for}\: \left|{\sqrt{x - 2}}\right| > 3 \\- \frac{x}{2} - 3 \sqrt{x - 2} + \left(x - 2\right) \log{\left(1 - \frac{\sqrt{x - 2}}{3} \right)} + {G_{2, 2}^{1, 1}\left(\begin{matrix} 1 & 2 \\1 & 0 \end{matrix} \middle| {x - 2} \right)} \log{\left(3 \right)} + 3 i \pi {G_{2, 2}^{1, 1}\left(\begin{matrix} 1 & 2 \\1 & 0 \end{matrix} \middle| {x - 2} \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} 2, 1 & \\ & 1, 0 \end{matrix} \middle| {x - 2} \right)} \log{\left(3 \right)} + 3 i \pi {G_{2, 2}^{0, 2}\left(\begin{matrix} 2, 1 & \\ & 1, 0 \end{matrix} \middle| {x - 2} \right)} - 9 \log{\left(1 - \frac{\sqrt{x - 2}}{3} \right)} + 1 & \text{otherwise} \end{cases} \]Sign Table
Variation Table
Plot
Elapsed Time: 0.0593 seconds