Browsing the web, I found quite a few integral representations for $\zeta(s)$ that use the Fractional part {x} or the Floor-function $\lfloor x\rfloor$ e.g.:
$$\zeta(s) = \dfrac{s}{s-1} - \frac12+s \int_1^\infty \frac{1/2-\{x\}}{x^{s+1}}\,\mathrm{d} x \qquad \Re(s)\gt 0\qquad(1)$$
or
$$\zeta(s) = \dfrac{s}{s-1} - s \int_1^\infty \frac{\{x\}}{x^{s+1}}\,\mathrm{d} x \qquad \Re(s)\gt 0\qquad(2)$$
Numerical evidence also suggests that the following expression holds:
$$\zeta(s) = \dfrac{1}{s-1} + \int_1^\infty \left(\frac{1}{\lfloor x \rfloor^{s}}-\frac{1}{x^s}\right)\,\mathrm{d} x \qquad \Re(s)\gt 0 \qquad(3)$$
That also leads to the very simple (trivial) integral:
$$\zeta(s) = \int_1^\infty \frac{1}{\lfloor x \rfloor^{s}}\,\mathrm{d} x \qquad \Re(s)\gt 1$$
Is there a way to derive (3) from (1) or (2)?
Thanks.