Function Expression :
\[f(x)=\frac{2x^3+2x^2\sqrt{x^2+1}+x-2}{2x} \]Domain
\[\left]-\infty, 0\right[ \cup \left]0, \infty\right[ \]Limits
\[\lim_{x \rightarrow-\infty}f(x) = 0 \]\[\lim_{x \overset{<}{\rightarrow0} }f(x) = +\infty \]
\[\lim_{x \overset{>}{\rightarrow0} }f(x) = -\infty \]
\[\lim_{x \rightarrow+\infty}f(x) = +\infty \]
\[ \]
Derivate
\[f^{\,\prime}(x)=\frac{1}{2 x} \left(\frac{2 x^{3}}{\sqrt{x^{2} + 1}} + 6 x^{2} + 4 x \sqrt{x^{2} + 1} + 1\right) - \frac{2 x^{3} + 2 x^{2} \sqrt{x^{2} + 1} + x - 2}{2 x^{2}} \]\[f^{\,\prime}(x)=\frac{2 x^{2}}{\sqrt{x^{2} + 1}} + 2 x + \frac{1}{\sqrt{x^{2} + 1}} + \frac{1}{x^{2}} \]
\[f^{\,\prime}(x)=\frac{2 x^{4} + 2 x^{3} \sqrt{x^{2} + 1} + x^{2} + \sqrt{x^{2} + 1}}{x^{2} \sqrt{x^{2} + 1}} \]
Integral
\[F(x) = \frac{x^{3}}{3} + \frac{x^{2} \sqrt{x^{2} + 1}}{3} + \frac{x}{2} + \frac{\sqrt{x^{2} + 1}}{3} - \log{\left(x \right)} \]Sign Table
Variation Table
Plot
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