Question 5.2.2: Compute the Laplace transform of f (t ) = 1.

From the definition, we observe that

L[1] =\int_{0}^{∞}{e^{−st} dt}=\underset{r\rightarrow \infty }{\lim }−\frac {1}{s}e^{−st}\overset{r}{\underset{0}{\mid }}=\underset{r\rightarrow \infty }{\lim }[−\frac {1}{s}e^{−sr} + \frac {1}{s}]=\frac {1}{s}

since e^{−sr} →0 as r→∞.