$$\mathcal{L}(U(t-5)e^{t-5} )$$
I’ve been taught this method and im not sure of and whether it will be correct, so I came here to ask.
$\mathcal{L}(U(t-5)e^{t-5} = \mathcal{L} (U(t-a) f(t-a) ) = e^{-as}F(s) $
a = 5, Therefore, $f(t-5) = e^{t-5} $
since $\mathcal{L} (f(t)) = F(s) $
$f(t) = e^t $
and $\mathcal{L} (f(t)) = \frac{1}{s-1} $
Therefore the answer is - $e^{-5s} \frac{1}{s-1} $
Am I correct ?
Yes, you have applied $$\mathcal{L}(U(t-5)e^{t-5} = \mathcal{L} (U(t-a) f(t-a) ) = e^{-as}F(s) = e^{-5s} \frac{1}{s-1}$$
correctly to get your answer $e^{-5s} \frac{1}{s-1}$