Chapter 5.2, Problem 11P

(a) Let $F\left(s\right)=L\left\{f\left(t\right)\right\}$, where $f\left(t\right)$ is piecewise continuous and of exponential order on $[0,\infty )$. Show that

$L\left\{{\displaystyle {\int }_0^{t}f\left(\tau \right)d\tau }\right\}=\frac{1}{s}F\left(s\right)$         (i)

Hint: Let $g_1\left(t\right)={\displaystyle {\int }_0^{t}f\left(t_1\right)d{t}_1}$ and note that $g_1\text{'}\left(t\right)=f\left(t\right)$. Then use Theorem $\mathrm{5.2.2}$.

(b) Show that for $n\ge 2$,

$L\left\{{\displaystyle {\int }_0^t{\displaystyle {\int }_0^{t_n}\cdots {\displaystyle {\int }_0^{t_2}f\left(t_1\right)d{t}_1}}}\cdots d{t}_n\right\}=\frac{1}{s^n}F\left(s\right)$         (ii)

Theorem $\mathrm{5.2.2}$: Suppose that $f$ is continuous and $f\text{'}$ is piecewise continuous on any interval $0\le t\le A$. Suppose further that $f$ and $f\text{'}$ are of exponential order. Then $L\left\{f\text{'}\left(t\right)\right\}$ exists for $s>a$, and moreover $L\left\{f\text{'}\left(t\right)\right\}=sL\left\{f\left(t\right)\right\}-f\left(0\right)$.