Wikipedia's article for the Dirichlet eta function tell us from the section Integral representations what is the representation due to Lindelöf, and what is the representation for $(s-1)\zeta(s)$ due to Jensen (I am saying the third and fourth paragraphs in the section).
Question. I am interested in the domain $0<\Re s<1$, where $s$ is the complex variable. In some of previous identities, is justified the derivation with respect the complex variable $s$ inside the integral sign? Can you provide such example, the related calculations to get the derivation and what is a rigurous justification (using Jensen's representation or Lindelöf's representation or a closely variation of this kind of integrals, I am saying including a similar complex power of the factor $1/2+it$ in the integrand)? Many thanks.
There is a derivation of $\displaystyle\eta(s) = \int_{-\infty}^\infty \frac{(1/2 + i t)^{-s}}{e^{\pi t}+e^{-\pi t}} \, dt$ (using the residue theorem)
Integral and Series Representations of Riemann's Zeta Function and Dirichlet's Eta Function and a Medley of Related Results