Let $a,b$ be real numbers, we consider the integral in form $\int\limits_{-\infty}^\infty \dfrac{|x|^a}{(1+x^2)^b}dx$. I know that when $a=0$, then the integral finite iff $b>1/2$.
I want to find a necessary and sufficient conditions on $a,b$ for which $\int\limits_{-\infty}^\infty \dfrac{|x|^a}{(1+x^2)^b}dx$ is finite, but I have no idea. Could any one help me?
As $|x| \to \infty$, the integrand is asymptotic to $|x|^{a-2b}$, so what you need for convergence there is $a-2b < -1$.
As $|x| \to 0$, the integrand is asymptotic to $|x|^a$, so here you need $a > -1$.
BTW, the integral can be evaluated in closed form (under the above assumptions): the answer is $$ {\frac {\Gamma \left( -1/2+b-a/2 \right) \Gamma \left( a/2+1/2 \right) }{\Gamma \left( b \right) }} $$