Let $\displaystyle f\in C^{\infty}(\mathbb{R},\mathbb{R})$, and define $\displaystyle f_{n}:=f_{n-1}^{\prime}+f_{n-1}f$ on $\displaystyle \mathbb{R}$ for $\displaystyle n\in\mathbb{N}$ with $\displaystyle f_{0}\equiv1$ on $\displaystyle \mathbb{R}$.

Show that

$\displaystyle \exp\bigg\{\displaystyle\int_{a}^{b}f(u)\mathrm{d} u\bigg\}=\displaystyle\sum_{\ell=0}^{\infty}f_{\el l}(a)\frac{(b-a)^{\ell}}{\ell!}\quad\text{for}\ a,b\in\mathbb{R}.$

Remark. In the case of constant function, i.e., $\displaystyle f\equiv c/(b-a)$ on $\displaystyle \mathbb{R}$ (provided that $\displaystyle b>a$) for some $\displaystyle c\in\mathbb{R}$,

we can compute that $\displaystyle f_{k}\equiv \big(c/(b-a)\big)^{k}$ on $\displaystyle \mathbb{R}$ for all $\displaystyle k\in\mathbb{N}$, which turns out to be a well known result.

That is,

$\displaystyle \mathrm{e}^{c}=\displaystyle\sum_{\ell=0}^{\infty} \frac{c^{\ell}}{\ell!}.$