Say I have this function
$$ \int f(x) dx $$
I now change $f(x)$ with $g(y) = f(x) \frac{dx}{dy}$
$$ \int g(y)\frac{dy}{dx}dx = \int g(y) dy $$
So far, so good?
Say I have
$$ \int f(x) f(x) dx $$
With the same change I have
$$ \int g(y)\frac{dy}{dx} g(y)\frac{dy}{dx}dx = \int g(y)\frac{dy}{dx} g(y)dy $$
The conclusion is that the integral $\int f(x) f(x) dx$ is NOT coordinate invariant.
I have to say I am extremely surprised by this. Can anyone explain in an intuitive manner why this is the case?