rules for Laplace transform

If $\mathcal{L}\{f(t)\}=F(s)$ , then

For deriving these rules, we start from the definition of Laplace transform. In the first case, we shall use the notation $s\!-\!a=r$ :

\mathcal{L}\{e^{at}f(t)\}=\int_{0}^{\infty}\!e^{-st}e^{at}f(t)\,dt=\int_{0}^{% \infty}\!e^{-(s-a)t}f(t)\,dt=\int_{0}^{\infty}\!e^{-rt}f(t)\,dt=F(r)=F(s\!-\!a).

In the second case, we make the change of variable $\frac{t}{a}=u$ and later use the notation $sa=r$ :

\mathcal{L}\{f(\frac{t}{a})\}=\int_{0}^{\infty}\!e^{-st}f(\frac{t}{a})\,dt=a\!% \int_{0}^{\infty}\!e^{-sau}f(u)\,du=a\!\int_{0}^{\infty}\!e^{-ru}f(u)\,du=aF(r% )=a\,F(as).