Question: Find $\int_1^\infty \frac{\{x\}}{x(x+1)}dx,$ where $\{x\}$ means $x - \lfloor x \rfloor$.
I have attempted to split this into two integrals, namely $$\int_1^\infty \frac{x}{x(x+1)}dx - \int_1^\infty \frac{\lfloor x \rfloor}{x(x+1)}dx,$$ however did not get anywhere. I have also considered that the average value of $\{x\}$ is $\frac{1}{2}$ (I think).
With mostly just instincts and jotting down a few numbers, I obtained an answer of $\ln{\sqrt{2}}$, which seems reasonable since Desmos approximated the answer (when the upper limit is large) as 0.30… I am only 17 so try to keep notation at my level, though this shouldn't really involve anything too complicated.