Let $a_k$ be the integral. Then
$$\sum_{k \ge 0} \frac{a_k}{k!} t^k = \int_1^{\infty} \frac{ \{ u \} }{u^2} e^{t \log u} \, du = \int_1^{\infty} \{ u \} u^{t-2} \, du = \frac{1 - \zeta(1 - t) - \frac{1}{t}}{1 - t}.$$$$\begin{eqnarray*} \sum_{k \ge 0} \frac{a_k}{k!} t^k &=& \int_1^{\infty} \frac{ \{ u \} }{u^2} e^{t \log u} \, du \\\ &=& \int_1^{\infty} \{ u \} u^{t-2} \, du \\\ &=& \frac{1 - \zeta(1 - t) - \frac{1}{t}}{1 - t} \\\ &=& \frac{1}{1 - t} \left( 1 - \sum_{n \ge 0} \frac{\gamma_n}{n!} t^n \right). \end{eqnarray*}$$
(Generating functions are good for more than combinatorics!) This is equivalent to Julian Rosen's answer, but (I think) packaged slightly more conveniently.