In Theorem 2.7 of Montgomery and Vaughan's Multiplicative Number Theory, they show that a constant arising in an earlier calculation is precisely the Euler-Mascheroni constant. While I understand the overall proof, I am having trouble with certain bounds they establish toward the end. Specifically, they derive that$$\delta \int_1^\infty \frac{x^{-1-\delta}}{\log 2x} \, dx \ll \delta + \delta \int_2^{e^{1/\delta}} \frac{dx}{x \log x} + \delta^2 \int_{e^{1/\delta}}^\infty x^{-1-\delta} \, dx \ll \delta \log \frac{1}{\delta}.$$It appears that the integral was split into three regions: $[1,2]$, $[2, e^{1/\delta}]$, and $[e^{1/\delta}, \infty)$. It is very likely that the bounds exploit the behavior of the integrand in each region, but I am struggling to reconstruct the details (I should also add that $\delta \rightarrow 0^{+}$).
Could someone explain how these bounds were obtained?