TITLE: Closed form for n-th anti-derivative of $\log x$
QUESTION [18 upvotes]: Is it possible to write a closed-form expression with free variables $x, n$ representing the n-th anti-derivative of $\log x$?

REPLY [6 votes]: An intuitive way to get the formula from the answer of Vladimir Reshetnikov without induction:
Recall that $\displaystyle\ln x=\lim_{s\rightarrow 0}\frac{x^s-1}{s}$. But the $n$th antiderivative of $x^s-1$ is very easy to compute:
$$\frac{1}{s}\underbrace{\int\ldots\int}_{n\;\text{times}}(x^s-1)dx=\frac{x^{s+n}}{s(s+1)\ldots(s+n)}-\frac{x^n}{s\cdot n!}=_{s\rightarrow0}\frac{x^n}{n!}\left(\ln x-\sum_{k=1}^n\frac{1}{k}\right).$$