Is this Riemann Zeta function integral formula known about?

Question

I discovered that $$\zeta(s)=\int_0^1\frac{(-\log(1-x))^{s-1}}{x(s-1)!}dx.$$ Is this an obvious result that is not worth much interest or is this new and unique?

Answer 1

This is, sadly, already known. Let $x\mapsto1-e^{-t}$, then $x\mapsto x/k$:

$$\begin{align}\int_0^1\frac{(-\log(1-x))^s}x~\mathrm dx&=\int_0^\infty\frac{x^s}{e^x-1}~\mathrm dx\\&=\int_0^\infty x^se^{-x}\left(\frac1{1-e^{-x}}\right)~\mathrm dx\\&=\int_0^\infty x^se^{-x}\sum_{k=0}^\infty e^{-kx}~\mathrm dx\\&=\sum_{k=1}^\infty\int_0^\infty x^se^{-kx}~\mathrm dx\\&=\sum_{k=1}^\infty\frac1{k^{s+1}}\int_0^\infty x^se^{-x}~\mathrm dx\\&=\zeta(s+1)\Gamma(s+1)\end{align}$$

As seen here.