In my notes I have the following two results. I recognise the latter from normal integration by substitution but it the first I don’t know where the derivative $|\varphi’(x)|$ has gone.
Why does the derivative of the substitution not appear in the first result? 
The first one is a very simple formula just based on the definition of $\mu$ in terms of $\phi$ and $\nu$ (and simple function approximation). No smoothness and derivatives are involved in it.