Question

Laplace Transform of
(1-e^-t)u(t-3)

Answer 1

\[ (1-e^{-t})u(t-3)\]

Answer 2

@Loser66 Lol, I was going to ask you.

Answer 3

I forgot all the stuff. ha!!

Answer 4

woah... you post the whole book ??

Answer 5

How are you in high school!

Answer 6

I'm not remembering the specific identity with multiplying by the heaviside step function, but by definition u(t - 3) is neutralizing any values below t < 3.
So the Laplace transform of (1 - e^(-t)) u(t - 3) is equal to:
\( \displaystyle \int_{0}^{\infty} (1 - e^{-t} ) u(t - 3) e^{-st} \ dt \)
= \( \displaystyle \int_{\color{blue}{3}}^{\infty} (1 - e^{-t} ) e^{-st} \ dt \)
We would need to substitute u = t - 3 so that the adjustment brings back our lower boundary of 0, which is the definition of the Laplace transform of our function after substitution