Question 4.4.7: Find the Laplace transform of the periodic function shown in......

\begin{aligned} \mathscr{L}\{E(t)\} & =\frac{1}{1-e^{-2 s}} \int_{0}^{2} e^{-s t} E(t) d t=\frac{1}{1-e^{-s t}}\left[\int_{0}^{1} e^{-s t} \cdot 1 d t+\int_{1}^{2} e^{-s t} \cdot 0 d t\right] \\ & =\frac{1}{1-e^{-2 s}} \frac{1-e^{-s}}{s} \quad \leftarrow 1-e^{-2 s}=\left(1+e^{-s}\right)\left(1-e^{-s}\right) \\ & =\frac{1}{s\left(1+e^{-s}\right)} . & (15) \end{aligned}