I need to expand the conditional variance of continuous random variable as a sum of integrals. Here is my try:
$$D(Y|X)=E[Y^2|X] - [E(Y|X)]^2 + EY = \int_y y^2f(x|y)dy -\left(\int_y yf(x|y)\right)^2dy $$ But I know that the second term is incorrect / could be rewritten otherwise. Maybe somoene has an insight?
Your formula of conditional variance is wrong...the correct one is the following
$$\mathbb{V}[Y|X=x]=\mathbb{E}[Y^2|X=x]-\mathbb{E}^2[Y|X=x]=$$
$$=\int_{-\infty}^{+\infty}y^2f_{Y|X}(y|x)dy-\left[\int_{-\infty}^{+\infty}yf_{Y|X}(y|x)dy \right]^2$$
There is nothing else to expand...