I am trying to evaluate the integral $$ I_k(x)=\int_1^x \log^k t \frac{\sqrt{t-1}}{t^2} dt $$ with $x$ tending to infinity.

In fact, I wish to have an estimate $$ \sum_{k=0}^\infty \frac{1}{\log^k x} \int_1^x \log^k t \frac{\sqrt{t-1}}{t^2} dt = C+E(x), $$ where $C$ is explicit and $E(x) \to 0$ as $x \to \infty$.

The integral inside may be actually rewritten as $$ I_k(x) = \int_1^x \log^k t \sqrt{1-\frac{1}{t}} \frac{dt}{t^{3/2}} = c_k \int_{1/x^2}^1 \sqrt{1-t^2} \log^k t dt, $$ hence can be possibly attacked via multiple zeta values.

Is that possible?