Proof: Expected value of the sum is equal to the sum...

I need to prove that

E[X+Y]=E[X]+E[Y] for two different cases.

Case 1: X and Y are independent.

Case 2: X and Y are not independent (general case).

I know I can take the double integral of (x+y)f(x,y)dxdy and split it into two, and the resulting integrals are the expected values of E[X] and E[Y], respectively, but I do not know if this is a proper proof of either case.

Thanks for your help.

Although you seem to be assuming continuous distributions, it isn't necessary. Prove the general case, then it works for independence as well, since you are simply factoring the joint distribution into the product of the marginals (and integrating over the whole space of one of them is easy when you are integrating a constant).