Answer to Question #343128 in Differential Equations for Tobias Felix

Question

Find the Laplace transforms of the following function:

10. L{e^-t}

Answer 1

$$F(s)=L(e^{-t})=\displaystyle\int_{0}^{\infin}e^{-st}e^{-t}dt$$

$$=\lim\limits_{A\to\infin}\displaystyle\int_{0}^{A}e^{-(s+1)t}dt=[-\dfrac{e^{-(s+1)t}}{s+1}]\begin{matrix} A \\ 0 \end{matrix}$$

$$=-0+\dfrac{1}{s+1}=\dfrac{1}{s+1}$$

$$L(e^{-t})=\dfrac{1}{s+1}$$