How to find integral $\int e^{-2x} /x\,dx\,$?
The question I am trying to answer is: If X, Y are random variables with joint pdf $f(x,y)=2e^{-2x} /x$ for $0\lt y \leq x$ what's the Covariance of X and Y? I get that the $cov(X,Y) = E(X*Y)-EX*EY$ and the E(X*Y) and EX are easy to get, but for the EY, I am first margining y by
$\int^\infty_y 2e^{-2x} /x dx$
But how in the world do you evaluate that integral? The complexity in solutions (which involve the exponential integral: https://en.wikipedia.org/wiki/Exponential_integral) that I've found suggest to me that I did not get the up correctly? If that looks correct, then how do you evaluate the integral?
You need not evaluate that integral. $EY=\int_0^{\infty} y\int_y^{\infty} 2e^{-2x}/x dx$. Interchange the integrals to get $EY=\int_0^{\infty}(\int_0^{x} ydy)2e^{-2x}/x dx$. This becomes $\int_0^{\infty} xe^{-2x}dx$ and you can evaluate this by a simple integration by parts.