Questions, can we then define \[\frac{d}{d\mathbb{I}}\sum_k ... \\ \int_a^b f(x) d\mathbb{I} \\ \sum_{\mathbb{I}} f_{\mathbb{I}}\]

Does there exist an integral that equals \(\zeta(s)\)... \[\zeta(s)\Gamma(s)=\int_0^\infty \frac{x^{s-1}}{e^x-1} \; dx \\ \frac{d}{dx}\frac{x^{s-1}}{e^x-1} = \frac{(e^x (s-x-1)-s+1) x^{s-2}}{(e^x-1)^2} \\\]

Therefor by the theory \[-\zeta(s)\Gamma(s)=\int_0^\infty \frac{(e^x (s-x-1)-s+1) x^{s-1}}{(e^x-1)^2} \; dx \\\]