General formula for integrating analytic function with linear term
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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 $$int_0^c t , f(t) , dt = cA - int_0^c int_0^t , f(tau) , dtau ,dt ; ,$$ 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.
integration definite-integrals analytic-functions
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