One more note. I got a really strange answer and I'm not sure if I'm doing it right. As i said,

E(X) = integral E(X|Y=y)Fy(y)

So in order to find E(X|Y) this is what i did

E(X|Y) = integral x fx|y(x|y) dx

now i used integration by parts here, and set dv to fx|y(x|y) dx, that means that v would be 1 since that is a gamma function and would integrate to 1 wouldn't it?

in which case u = x and du = dx

After simplifying i get E(X|Y) as...0 which results in E(X) = 0 as well...