The function $q$ associates to any real $x$ the real number $q ( x ) = x - \lfloor x \rfloor - \frac { 1 } { 2 }$, where $\lfloor x \rfloor$ denotes the integer part of $x$.
Show that $\int _ { 1 } ^ { + \infty } \frac { q ( u ) } { e ^ { t u } - 1 } \mathrm {~d} u$ is well-defined for all real $t > 0$.