$\def\d{\mathrm{d}}$After I was studying variations of the integral representation for $\zeta(3)$ due to Beukers, see the section More complicated formulas from this Wikipedia, I've thought an exercise. The motivation is this antiderivative $$-\int_0^1e^y\frac{\log(xy)}{1-xy}\,\d x=-\frac{e^y}{y}\operatorname{Li}_2(1-xy)+\mathrm{constant}.$$ For real numbers $x\geq 2$, $$f(x):=-\int_0^1\frac{e^y}{y}(\operatorname{Li}_x(1-y)-\zeta(x)) \,\d y,$$ I've calculated with Wolfram Alpha online calculator some particular values at integer values $x=n\geq 2$.
Question.
A) Is it possible to prove that $f(x)$ is decreasing for $x\geq 2$?
B) Is it possible to calculate $$\lim_{x\to\infty}f(x)?$$ Thanks in advance.
Also I know Euler's Zeta function $\zeta(x)$ tends to $1$ as $x\to\infty$. With respect the definite integral I believe that is (impossible) difficult to get a closed-form for such values.