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I am looking for a simple way to show that the mean of the Cauchy distribution us undefined. This is because this integral diverges:

[itex]\underset{-\infty}{\overset{\infty}{\int}}\frac{x}{x^{2}+a^{2}}dx[/itex]Now, I know one proof which replaces the limits of integration with -x1 and x2. After carrying our the definite integration we are left with [itex]\frac{1}{2}\ln\left(\frac{a^{2}+x_{2}^{2}}{a^{2}+x_{1}^{2}}\right)[/itex] and finally (by Taylor Series expansion) [itex]2\ln x_{2}-2\ln x_{1}+smaller terms[/itex] . Then allowing x1 and x2 to approach infinity shows that the intergral diverges.

My question is now: is it sufficient, on any level, just to look at the antiderivative [itex]\ln(a^{2}+x^{2})[/itex], state that it is an increasing function of x, and simply conclude from it that the integral diverges ?

Thanks

LR

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# To show that an integral is divergent

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