To show that an integral is divergent

[longrob]

Hi all

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

$\underset{-\infty}{\overset{\infty}{\int}}\frac{x}{x^{2}+a^{2}}dx$​

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 $\frac{1}{2}\ln\left(\frac{a^{2}+x_{2}^{2}}{a^{2}+x_{1}^{2}}\right)$ and finally (by Taylor Series expansion) $2\ln x_{2}-2\ln x_{1}+smaller terms$ . 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 $\ln(a^{2}+x^{2})$, state that it is an increasing function of x, and simply conclude from it that the integral diverges ?

Thanks
LR

 

[Office_Shredder]

The antiderivative increasing is definitely not enough. If you're integrating ANY positive function you'll get an increasing antiderivative, but there are plenty of such functions which are integrable

 

[longrob]

Thanks for your reply. That's what i thought. So I assume the easiest/typical way is the way I described above ?