What is $\int_0^\infty \int_x^\infty \frac{1}{y} e^{-y/2} \, dy \, dx$?

Question

Frankly I don't know how to simplify or which theorem to apply. I do know theorem on $e^{-y^2}$ (gaussian) but I don't think I can manipulate the current question to that form?

P.S Answer is 2.

Answer 1

Changing the order of integration yields

$$\int_0^\infty \int_x^\infty\frac1ye^{-y/2}\,dy\,dx=\int_0^\infty \int_0^y \frac1ye^{-y/2}\,dx\,dy$$

Can you finish now?