Find the antiderivative of a function with a finite series and factorials

Question

If $n\in\mathbb{N},s\leq n$, I know that $$ \int_0^1 t^s(1-t)^{n-s-1}dt=\frac{s!(n-s-1)!}{n!}. $$ I would like to find a similar formula: is there a function $f(t)$ such that $$ \int_0^1 f(t) dt=\frac{s!(n-s-1)!}{n!}\sum_{k=s+1}^n\frac{1}{k} $$ and how can I find it?

Answer 1

An obvious answer would be $$f(t) = \sum_{k=s+1}^n \dfrac{t^s (1-t)^{n-s-1}}{k}$$ Is there some reason that is unsuitable?

Answer 2

One more obvious ( see @Robert Israel answer ) is a "${\it\mbox{constant function}}$":

$$ {\rm f}\left(t\right) \equiv \frac{s!(n-s-1)!}{n!}\sum_{k=s+1}^n\frac{1}{k} $$