I am trying to solve the following:
$\int_0^\infty{x^{-2}p(x)}dx$
Where $p(x)$ is the inverse gamma pdf:
$p(x)=\frac{\beta^\alpha}{\Gamma(\alpha)}(1/x)^{\alpha+1} exp(-\beta/x) $
Numerically, I can see that the solution exists when I do Monte Carlo analysis over 100K scenarios, but I just don't have the mathematical background to solve the equation. Help is much appreciated!