Function Expression :
\[f(x)=ln(x+\frac{1}{x} )+\frac{x^2-1}{x^2+1} \]Domain
\[\left]0, \infty\right[ \]Limits
\[\lim_{x \overset{>}{\rightarrow0} }f(x) = +\infty \]\[\lim_{x \rightarrow+\infty}f(x) = +\infty \]
\[ \]
Derivate
\[f^{\,\prime}(x)=- \frac{2 x \left(x^{2} - 1\right)}{\left(x^{2} + 1\right)^{2}} + \frac{2 x}{x^{2} + 1} + \frac{1 - \frac{1}{x^{2}}}{x + 1 \cdot \frac{1}{x}} \]\[f^{\,\prime}(x)=\frac{x^{4} + 4 x^{2} - 1}{x^{5} + 2 x^{3} + x} \]
\[f^{\,\prime}(x)=\frac{x^{4} + 4 x^{2} - 1}{x \left(x^{4} + 2 x^{2} + 1\right)} \]
Integral
\[F(x) = x \log{\left(x + \frac{1}{x} \right)} \]Sign Table
Variation Table
Plot
Elapsed Time: 0.0040 seconds