Laplace Transform with translation theorem?

Question

How do I solve the following Laplace transform $\mathcal{L}\{te^{t-5}\mathcal{U}(t-5)\}(s)$ where $\mathcal{U}$ is the unit step?

Answer 1

The integral you use is

$$\begin{align}\int_5^{\infty} dt \: t \, e^{(t-5)} e^{-s t} = \underbrace{e^{-5 s}\int_0^{\infty} dt \: (t+5) e^{-(s-1) t}}_{t+5 \leftarrow t} \end{align}$$

Can you take it from here?