The Law of the Unconscious Statistician

In the section on variance, the identity \mathbb E(X^2) = \int x^2 f(x) dx (and similar identities) are used without comment. Since X^2 is a random variable in its own right, \mathbb E(X^2) has a different definition: \int x g(x) dx where g(x) is the PDF of X^2. These two expressions, although different, calculate the same thing by the Law of the Unconscious Statistician in the special case of the function t \mapsto t^2.

I think the book should remark on this, although I suspect an actual proof of the law is beyond the scope of the book.