Given an analytic function $\, f : \mathbb{R} \rightarrow \mathbb{R}$ with no known closed formula for its antiderivative. Assume further that by some clever tricks you managed to calculate the definite integral for a fixed number $c$: $$A := \int_0^c f(t) \,dt \; .$$ Is there a way to calculate $\int_0^c t \, f(t) \, dt$ in terms of $A$, $f$ and $f^{(n)}$?
Integration by parts yields $$\int_0^c t \, f(t) \, dt = cA - \int_0^c \int_0^t \, f(\tau) \, d\tau \,dt \; ,$$ which made it seem plausible to me that such a formula exists.
Related Question:
This problem is an abstraction of my more concrete question asked here.
