Chapter 7.8, Problem 65ES

A transform is a formula that converts or “transforms� one function into another. Transforms are used in applications to convert a difficult problem into an easier problem Whose solution can then be used to solve that original difficult problem. The Laplace transform of a function $f(t),$ which plays an important role in the study of differential equations, is denoted by $L\{f(t)\}$ and is defined by

$L\{f(t)\}={\displaystyle {\int }_0^{+\infty }{e}^{-st}f(t)dt}$

In this formula $s.$ is treated as a constant in the integration process; thus, the Laplace transform has the effect of transforming $f(t)$ into a function of $s.$ Use this formula in these exercises.

In each part, find the Laplace transform.

$\begin{array}{l}(a)f\left(t\right)=t,s>0\\ \left(b\right)f(t)=t^2,s>0\\ \left(c\right)f(t)=\left\{\begin{array}{cc}0,& t<3\\ 1,& t\ge 3\end{array},s>0\right.\end{array}$