$\mathcal{L} ( (t+5)U(t-1) ) $

Question

$\mathcal{L} ( (t+5)U(t-1) ) = U(t-a)f(t-a) = e^{-as} F(s) $

a= 1

I am having trouble dealing with $f(t-a)$ and finding $F(s)$

$f(t-1) = e^{t-3} = e^{(t-1)-3+1} = e^{(t-1)-2} $

$f(t) = e^{t-2} $

How do I Laplace transform this ?

Answer 1

Hint:

$$ \mathcal{L}\left[ (t+5)U(t-1)\right]= \mathcal{L} \left[(t-1+6)U(t-1)\right] = \mathcal{L}\left[ (t-1)U(t-1) + 6U(t-1)\right] = \mathcal{L} \left[(t-1)U(t-1)\right] + 6\mathcal{L}\left[ U(t-1)\right]$$

Where we know: $$ \mathcal{L}U(t) = \frac{1}{s},$$ and $$\mathcal{L}tU(t)=\frac{1}{s^2}.$$