So let $g:[a,b]\rightarrow \mathbb{R}$ be a $C^{n+1}$ function with $n\geq 0$. Suppose $a\leq x \leq b$, and let $h = x-a$.
I want to show by changing variables in the fundamental theorem that:
$g(x)=g(a)+h\int_0^1g^\prime(a+th)dt$ Where I'm using the following equivalent version of the fundamental theorem:
$g(x)=g(a)+\int_a^xg^\prime(t)dt$
Naturally, the change of variables here is $\phi(t)=a+th$.
But I'm having a bit of trouble applying this change to the bounds. Namely, I'm having trouble dealing with all possible values of $h$ and what they're telling me.
This problem is presented among the more ambient task of deriving taylor's theorem with integral remainder. (Hence the $C^{n+1}$ hypothesis.)