Evaluation of integral using laplace transform

Question

How to evaluate the integral $\displaystyle\int_0^\infty\frac{e^{-at}-e^{-bt}}{t}dt$? As this is Laplace transform of $\frac{1}{t}$ at $s=a$, I tried with division by $t$ property, i.e. division by $t$ and integration from $s$ to infinity, but I am getting stuck at diverging integral of $\log s$ from $0$ to $\infty$. (Laplace transform of $\frac{1}{t}$ is $\int_0^\infty \frac{1}{s}ds=\log s$.

Please help

Answer 1

\begin{eqnarray} \int_0^\infty\frac{e^{-at}-e^{-bt}}{t}\,dt&=&\int_0^\infty\left(\int_a^be^{-st}\,ds\right)\,dt=\int_a^b\left(\int_0^\infty e^{-st}\,dt\right)\,ds\\ &=&\int_a^b\frac{1}{s}\,ds=\ln\frac{b}{a}. \end{eqnarray}