Laplace transform of $f(\frac{t}{c})$. So in order for me to able to do the Laplace transform of the following function, what I did was the following:
$L(f(\frac{t}{c})) = \int_{0}^{\infty}e^{-st}f(\frac{t}{c})dt$
Now because I don't know of an easier way to solve this, what I did was integration by parts(feel free to correct me if there is an easier way to do this other than integration by parts)
Anyway, then I said:
Letting $dv = e^{-st}dt \implies v = \frac{-1}{s}e^{-st}$ and $u = f(\frac{t}{c}) \implies du = \frac{1}{c} f(\frac{t}{c})$
But what to do now? Am I even going in the right direction? The algebra gets considerably more mucky after this, and I do not know of a better way to solve it.
$$L \left \{f\left(\frac{t}{c}\right)\right \}= \int_{0}^{\infty}e^{-st}f(\frac{t}{c})dt$$ Substitute $t=uc \implies dt=cdu$: $$L \left\{f\left(\frac{t}{c}\right)\right\} =c \int_{0}^{\infty}e^{-(sc)u}f(u)du=cF(sc)$$