Definition of Laplace - change of variables

Question

Any hint? I am not really understanding where to start

Answer 1

$$\begin{array}{rcl} \mathcal L \{f(at)\}(s) &=& \displaystyle \int_0^\infty e^{-st}f(at) \ \mathrm dt \\ &=& \displaystyle \int_0^\infty e^{-su/a}f(u) \ \mathrm d(u/a) \\ &=& \displaystyle \int_0^\infty e^{-st/a}f(t) \ \mathrm d(t/a) \\ &=& \displaystyle \dfrac1a \int_0^\infty e^{-(s/a)t}f(t) \ \mathrm dt \\ &=& \dfrac1a \mathcal L \{f(t)\}\left(\dfrac sa\right) \\ &=& \dfrac1a F\left(\dfrac sa\right) \\ \end{array}$$

The substitution used is $u=at$, where $\mathrm du = a \ \mathrm dt$.