"My E(x) comes to be approaching to infinity as x approaches infinity."
What does it mean?
If E[X] is the expected value of X, what is x? the dummy variable?
Once you have taken the expected value, the dummy variable x is integrated
out. It should be independent from your x.
Moreover, since the joint p.d.f. is a product of the functions of the dummy
variables x and y, you should know that X, Y are independent, and X should
have an exponential distribution.