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 $$\int_1^\infty \frac{x}{x(x+1)} - \int_1^\infty \frac{\lfloor x \rfloor}{x(x+1)},$$ however did not get anywhere significant. I have als...