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Understanding Laplace Transform of a piece-wise defined function

Let’s say $f(t)$ is a positive, even and continuous function. Further, let’s say that for all $t\in [0, 1]$, $f(t) = 0$; For all $t \in [1,2]$, $f(t) = 0.5$; For all $t\in [2,3]$, $f(t) = 0.25$; For all $t\in [3,4]$, $f(t) = 0.75$; For all $t \in [4,5]$, $f(t) = 0.25$; For all $t\in [5,6]$, $f(t) = 0$; For all $t\in [6,7]$, $f(t) = 0.5$; and for all $t \in [7,\infty]$, $f(t) = 1$.
I’m trying to understand the Laplace Transform of this function. I know how to do it for $f(t) = a$ for some constant $a$, but what’s the best way to go about doing this?

A:

The Laplace Transform of the piecewise function $f$ is:
$$ \mathcal{L}(f) = \mathcal{L}(0,0) + \mathcal{L}(0.5,0) + \mathcal{L}(0.25,0) + \mathcal{L}(0.75,0) + \mathcal{L}(0.5,0) + \mathcal{L}(0,0) + \mathcal{L}(0,0) + \mathcal{L}(0.5,0) + \mathcal{L

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