\[ \begin{aligned} F(s) &= \int_{0}^{\infty} f(t)e^{-st}\,dt \\ &= \int_{0}^{\infty} t e^{-st}\,dt \\ &= \frac{t e^{-st}}{-s} \Bigg|_{0}^{\infty} - \int_{0}^{\infty} \frac{e^{-st}}{-s}\,dt \\ &= \left( \frac{\infty e^{-s\infty} - 0e^{-s0}}{-s} \right) - \frac{1}{-s}\int_{0}^{\infty} e^{-st}\,dt \\ &= (0 - 0) - \left( \frac{e^{-s\infty} - e^{-s0}}{s^2} \right) \\ &= \left( 0 - \frac{-1}{s^2} \right) \\ &= \frac{1}{s^2} \end{aligned} \]