How do you evaluate the sum
$$\sum_{k=1}^\infty\int_{\sqrt k}^{\sqrt{k+1}} \left(\frac{x^2}{k}-1\right)\, dx$$
without using $\zeta(2)$?
I can see some relationship to $\dfrac1{\lfloor x^2\rfloor}$ and furthermore with $\dfrac{x^2}{\lfloor{x^2}\rfloor}-1$, but I didn't make much progress this way either.