Law of total variance; integral expansion.

$$Var(Y) = E(Var(Y|X)) + Var(E(Y|X))$$

Here is an expansion:

$$int_X int_Y( f_X (x) (y^2 f(y|x) – mu^2_{Y|x})) dy dx + int_X int_Y( f_X(x) (y f(y|x)-2 y^2 f(y|x) + mu^2_Y)) dy dx $$

Is there a mistake / could it be rewritten differently?