expectation of cauchy distribution when power less than 1

Question

How do I prove that $E|X|^p$, p<1 is less than infinity when $X$ is Cauchy distributed?

The pdf is $f(x) = 1/\pi(1+x^2)$.

When $p=1$, EX integrates nicely to $\frac{1}{2\pi} \ln(1+x^2)$, but its not so nice when eg $p=1/2$. Then we get something like $\ln \left|2x+2^{1+\frac{1}{2}}x^{\frac{1}{2}}+2\right| - arctan(...)$

Answer 1

$E|X|^{p}=\int\frac {|x|^{p}} {\pi (1+x^{2})}dx\leq \int_{-1}^{1} \frac 1 {\pi} |x|^{p}dx+\frac 1{\pi} \int_{|x|>1} \frac 1 {|x|^{2-p}}dx<\infty$.