Matemáticas la transformada de laplace

Usemos la definición de la transformada de Laplace

$$\begin{align}&F(s)={\mathcal {L}}\left\{f(t)\right\}=\int _{0}^{\infty }e^{-st}f(t)\,dt\\&\\&F(s)=\int _{0}^{2 }e^{-st}f(t)\,dt +\int _{2}^{\infty }e^{-st}f(t)\,dt\\&\\&F(s)=4\int _{0}^{2 }e^{-st}\,dt \\&\\&F(s)=4\left.\left(-\frac{e^{-st}}{s}\right)\right|_{t=0}^{t=2}\\&\\&F(s)=4\left(-\frac{e^{-2s}}{s}+\frac{1}{s}\right)\\&\\&\huge\boxed{F(s)=\frac{4}{s}-\frac{4e^{-2s}}{s}}\end{align}$$