The last formula seems to be an integral expression for the Dirichlet $\eta$ function:
$$\eta(s)=\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k^s}\quad \Re(s) > 0$$
Using the relationship with the usual Riemann $\zeta$ function
$$\eta(s)=(1-2^{1-s})\zeta(s)$$
and this integral expression, you get that integral in the notebook:
$$\eta(s) = \frac1{\Gamma(s)} \int_0^{\infty}\frac{x^{s-1}}{e^x+1}\mathrm dx$$
There is also this closely related integral (which Arturo mentions in his answer).
The (first part of the) second line looks to be the chain of relations relating Riemann $\zeta$, Dirichlet $\eta$, and Dirichlet $\lambda$:
$$\frac{\zeta(s)}{2^s}=\frac{\lambda(s)}{2^s-1}=\frac{\eta(s)}{2^s-2}$$
In the second part, the expressions look to be the differentiation of Dirichlet $\eta$, but the screenshot is fuzzy around that region...