Change-of-variables with a product of two functions in the integrand

Question

How does the change-of-variable $x=\phi(t)$ formula $$ \int_{\phi(a)}^{\phi(b)} f(x) dx = \int_{a}^{b} f(\phi(t))\phi'(t)dt$$ look like when the integral looks like this: $$\int_{\phi(a)}^{\phi(b)} f(x) g(x) dx$$ I focussed on the product in the integrand, and tried using the product rule, as well as integration by parts, but both approaches led nowhere. Is this after all just $dx$ replaced by $\phi'(t)dt$ and then $\phi$ plugged in?, meaning $$\int_{a}^{b} f(\phi(t)) \, g(\phi(t)) \, \phi'(t)dt$$ This sure feels like textbook material, but I couldn't find it anywhere.