Determine the asymptotic behaviour of the following integral:
$(1)$ : $f(\epsilon)=f(0)+ \int^{\epsilon}_{0}f'(x) dx$ as $\epsilon \rightarrow 0$
I have in my notes that:
$(2)$
: $f(\epsilon)=f(0) + \left[(x-\epsilon)f'(x)\right]^{\epsilon}_{0} + \int^{\epsilon}_{0}(\epsilon - x)f''(x) dx$
Then
$f(\epsilon)=\sum^{N}_{n=0}\dfrac{\epsilon^{n}f^{n}(0)}{n!} + \dfrac{1}{N!}\int^{\epsilon}_{0}(\epsilon - x)^{N}f^{(n+1)}(x) dx$ (provided remainder is small)
I understand that we need to use integration by parts to obtain an asymptotic approximation however I dont understand how to go from $(1)$ to $(2)$