Function Expression :
\[f(x)=ln\frac{x^2-2}{2x-1} \]Domain
\[\left]- \sqrt{2}, \frac{1}{2}\right[ \cup \left]\sqrt{2}, \infty\right[ \]Limits
\[\lim_{x \overset{>}{\rightarrow- \sqrt{2}} }f(x) = -\infty \]\[\lim_{x \overset{<}{\rightarrow\frac{1}{2}} }f(x) = +\infty \]
\[\lim_{x \overset{>}{\rightarrow\sqrt{2}} }f(x) = -\infty \]
\[\lim_{x \rightarrow+\infty}f(x) = +\infty \]
\[ \]
Derivate
\[f^{\,\prime}(x)=\frac{\left(2 x - 1\right) \left(\frac{2 x}{2 x - 1} - \frac{2 \left(x^{2} - 2\right)}{\left(2 x - 1\right)^{2}}\right)}{x^{2} - 2} \]\[f^{\,\prime}(x)=\frac{2 \left(x^{2} - x + 2\right)}{2 x^{3} - x^{2} - 4 x + 2} \]
\[ \]
Integral
\[F(x) = \frac{21999074304 \sqrt{2} x \log{\left(\frac{x^{2}}{2 x - 1} - \frac{2}{2 x - 1} \right)}}{21999074304 \sqrt{2} + 31133992066} + \frac{31133992066 x \log{\left(\frac{x^{2}}{2 x - 1} - \frac{2}{2 x - 1} \right)}}{21999074304 \sqrt{2} + 31133992066} - \frac{31133992066 x}{21999074304 \sqrt{2} + 31133992066} - \frac{21999074304 \sqrt{2} x}{21999074304 \sqrt{2} + 31133992066} - \frac{20134454914 \sqrt{2} \log{\left(x - \sqrt{2} \right)}}{21999074304 \sqrt{2} + 31133992066} - \frac{28431152575 \log{\left(x - \sqrt{2} \right)}}{21999074304 \sqrt{2} + 31133992066} + \frac{59565144641 \log{\left(x + \sqrt{2} \right)}}{21999074304 \sqrt{2} + 31133992066} + \frac{42133529218 \sqrt{2} \log{\left(x + \sqrt{2} \right)}}{21999074304 \sqrt{2} + 31133992066} - \frac{15566996033 \log{\left(\frac{x^{2}}{2 x - 1} - \frac{2}{2 x - 1} \right)}}{21999074304 \sqrt{2} + 31133992066} - \frac{10999537152 \sqrt{2} \log{\left(\frac{x^{2}}{2 x - 1} - \frac{2}{2 x - 1} \right)}}{21999074304 \sqrt{2} + 31133992066} \]Sign Table
Variation Table
Plot
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