Chapter 5.5, Problem 25P

(a) If $f\left(t\right)=1-u_1\left(t\right)$, find $L\left\{f\left(t\right)\right\}$; compare with Problem 19. Sketch the graph of $y=f\left(t\right)$.

(b) Let $g\left(t\right)={\displaystyle \underset{0}{\overset{t}{\int }}f\left(\xi \right)}d\xi$, where the function $f$ is defined in part (a). Sketch the graph of $y=g\left(t\right)$ and find $L\left\{g\left(t\right)\right\}$.

(c) Let $h\left(t\right)=g\left(t\right)-u_1\left(t\right)g\left(t-1\right)$, where $g$ is defined in part (b). Sketch the graph of $y=h\left(t\right)$ and $L\left\{h\left(t\right)\right\}$.

$f\left(t\right)=\{\begin{array}{ll}1,\hfill & 0\le t<1\hfill \\ 0,\hfill & t\ge 1\hfill \end{array}$