\[ \begin{aligned} F(s) &= \int_{0}^{\infty} f(t)e^{-st}\,dt \\ &= \int_{0}^{\infty} ae^{-st}\,dt \\ &= a \int_{0}^{\infty} e^{-st}\,dt \\ &= a \left( \frac{e^{-st}}{-s} \right) \Big|_{t=0}^{t=\infty} \\ &= a \left( \frac{e^{-(s)(\infty)} - e^{0}}{-s} \right) \\ &= a \left( \frac{0 - 1}{-s} \right) \\ &= \frac{a}{s} \end{aligned} \]