Laplace Transform of exponential

Now let's go slightly more complicated:


$\begin{eqnarray*} \int\limits_{0}^{\infty}{e^{at}e^{-st}dt} &=& \int\limits_{0}^{\infty}{e^{(a-s)t}dt} \\ &=& \frac{{e^{(a-s)t}dt}}{(a-s)}|_{0}^{\infty} \\ &=& \frac{1}{s-a} \end{eqnarray*}$

Note that in the third equal sign above, we choose $s \lt a$ so that we $t \to \infty$, the exponential converge to $0$.