# Laplace Transform

## Homework Statement

Is there an easier way of solving this rather than doing the integral?
Find the laplace transform of:

$$t{e^{ - t}}u(t - 1)$$

## The Attempt at a Solution

$$\int_1^\infty {t{e^{ - t}}} {e^{ - st}}dt = \int_1^\infty {t{e^{ - t(1 + s)}}} dt = \frac{{{e^{ - (s + 1)}}}}{{{{(1 + s)}^2}}} + \frac{{{e^{ - (s + 1)}}}}{{1 + s}}$$

LCKurtz
Homework Helper
Gold Member

## Homework Statement

Is there an easier way of solving this rather than doing the integral?
Find the laplace transform of:

$$t{e^{ - t}}u(t - 1)$$

## The Attempt at a Solution

$$\int_1^\infty {t{e^{ - t}}} {e^{ - st}}dt = \int_1^\infty {t{e^{ - t(1 + s)}}} dt = \frac{{{e^{ - (s + 1)}}}}{{{{(1 + s)}^2}}} + \frac{{{e^{ - (s + 1)}}}}{{1 + s}}$$
]

That integral isn't that tough, but if you prefer, you can use some of the theorems. For example, you have, if ##\mathcal L(f(t)) = F(s)## then ##\mathcal L(tf(t))=-F'(s)##. Also, ##\mathcal L(e^{at}f(t) = F(s-a)## and ##\mathcal L(u(t-1))=\frac{e^{-s}}{s}##.

So starting with ##\mathcal L(u(t-1))=\frac{e^{-s}}{s}= F(s)##, then ##\mathcal Le^{-t}u(t-1)=F(s+1) =\frac{e^{-(s+1)}}{s+1}## and $$L(te^{-t}u(t-1)) = -\frac d {ds}\left(\frac{e^{-(s+1)}}{s+1}\right)=\frac{e^{-s+1}(s+1)+e^{-s+1}}{(s+1)^2}$$which is the same answer.

True - it isn't that tough. But I am not so keen on integration by parts if I can avoid it.
Thanks for the reply.