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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:


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.)

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Using the change of variables theorem take

$t= a + u(x-a)$ so that when $u=0$ then $t=a$ and when $u=1$ then $t=x$. Then we have that $dt=hdu$ and

$g(x) = g(a) + \int_a^x g^\prime(t)dt = g(a) + h\int_0^1 g^\prime(a + uh)du$.

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