# Laplace Transform of multiplied term like u(t)u(4−t)

kasibug1v 2022-10-05 Answered
Laplace Transform of multiplied term like $u\left(t\right)u\left(4-t\right)$
First of all,
If this is a two-terms function I'd be simple. It will produce
$\mathcal{L}\left[u\left(t\right)\right]=\frac{1}{s}$
Except, I'm not sure what to do with $u\left(4-t\right)$. If it was $u\left(t-4\right)$, it would be simpler.
Secondly,
Since these two terms are not separate. I couldn't find a property to help me deal with it.
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## Answers (1)

farbhas3t
Answered 2022-10-06 Author has 6 answers
I think the easiest way is to use the definition of the Laplace transform directly. Note that
$u\left(t\right)u\left(4-t\right)=\left\{\begin{array}{ll}1,& 0
so
$\mathcal{L}\left[u\left(t\right)u\left(4-t\right)\right]\left(s\right)={\int }_{0}^{4}{e}^{-st}\phantom{\rule{thinmathspace}{0ex}}dt,$
which I am sure you can work out.
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