Laplace transform using change of scale property

Question

If Laplace transform of $f(t)=\phi(s)$, then Laplace transform of $e^{bt}f(at)$ is

Answer 1

You are given: $$ \int_{t=0}^{\infty}f(t)e^{-st}\;dt=\phi (s) $$ and are asked for: $$ \int_{t=0}^{\infty} e^{bt}f(at)e^{-st}\;dt=\int_{t=0}^{\infty} f(at)e^{-(s-b)t}\;dt $$ Now put $t'=at$ when the integral becomes: $$ \int_{t=0}^{\infty} e^{bt}f(at)e^{-st}\;dt=\int_{t'=0}^{\infty}\frac{1}{a}f(t')e^{-\frac{s-b}{a}t'}\;dt'=\frac{1}{a}\phi\left(\frac{s-b}{a}\right) $$