Question

Find the laplace transform of f(t)={0,(t−4),0≤t<4t≥4

Answer 1

Presumably,


`f(t)=\begin{cases}0&\text{for }0\le t<4\\t-4&\text{for }t\ge4\end{cases}`

By definition of the Laplace transform,


`\mathscr L_s\{f(t)\}=\displaystyle\int_0^\infty f(t)e^(-st)\,\mathrm dt=\int_4^\infty(t-4)e^(-st)\,\mathrm dt`

Integrate by parts, taking


`u=t-4\implies\mathrm du=\mathrm dt`

`\mathrm dv=e^(-st)\,\mathrm dt\implies v=-\frac1se^(-st)`

so we get


`\displaystyle\mathscr L_s\{f(t)\}=\left(-\frac1se^(-st)(t-4)\right)\bigg|_(t=4)^(t\to\infty)+\frac1s\int_4^\infty e^(-st)\,\mathrm dt`

`=-\frac1{s^2}e^(-st)\bigg|_(t=4)^(t\to\infty)`

`=(e^(-4s))/(s^2)`