I'm an amateur so please go easy on me if this is a bit low-level. I'm working on a paper and derived the following which looks like it's just a restatement of Taylor's approximation. It's not exactly related to what I'm working on, but I'm curious if this is worth deriving in the actual paper or if the result is called something and should just be referenced by name instead.
$$ \int_{a}^{b} f(x) \,dx=\sum_{k=1}^{\infty}(b-a)^k \frac{f^{(k-1)}(a)}{k!} $$
What you have here is term-by-term integration of a power series. It is already known as you suspected.