no i dont think so. As your wrote it, your E(X|Y) would almost certainly be a function of X, so you know that it is incorrect. E(X|Y) should be a function of Y only (or a constant).

Lets start from the definition of expectation for any random variable, x:

Similarly for the conditional variable "X|Y"

You can make this substitution

to get:

is this what you meant?

is similar except the starting definition is

You try and finish. If you need further help, be sure to post the density functions you are supposed to be manipulating.