Integral at infinity

Question

Let us fix a parameter $s \in \left(\, 0,1\,\right)$, the problem is to compute, or at least study the convergence, the following integral $$ I = \int_{2}^{\infty}\left[\frac{1 - x^{s}}{\left(x - 1\right)^{1 + s}} + \frac{x^{s} + 1}{\left(1 + x\right)^{1 + s}}\right]\mathrm{d}x $$ It seems to be quite easy to prove that $$ I\leq C \int_{2}^{\infty}\frac{\mathrm{d}x}{x^{1 + s}} $$ for some positive constant $C > 0$, but the problem is to prove a lower bound.