A change of parametrization yields:
$$ \int f(x) dx \to \int f(y(x)) \frac{dy}{dx} dx = \int f(y) dy $$
But I cannot work out how to do it when dealing with a product of two functions. Intuitively, I expected the following:
$$ \int h(x)g(x) dx \to \int h(y(x)) \frac{dy}{dx} g(y(x)) \frac{dy}{dx} dx = \int h(y) g(y(x))\frac{dy}{dx} dy $$
But I cannot get rid of the dy/dx term.
Can anyone show how to do it?