Laplace Transform of a piecewise function

[loki4000]

So I am trying to do circuit analysis with sawtooth pulse (problem 10) and the textbook that I am working with is extremely uninformative on topic.

there is only one worked example 1.9 (see picture) and more confusing than helpful.

So I was wondering if anyone can explain me how to solve this kind of problems (see picture below).
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[HallsofIvy]

To answer your last question first, "F(s)" is the usual notation for the Laplace transform of "f(t)". Your text is using "U(s)" to indicate the Laplace transform of "u(s)". Your question "How do you find \(\displaystyle F(s)\) for \(\displaystyle U_2(s)\) makes no sense. They are just different notations for the same thing. Did you mean "How do you find \(\displaystyle u_2(x)\) for \(\displaystyle U_2(s)\)? That is, find the inverse Laplace transform?

You do know the definition of "Laplace transform", don't you?
\(\displaystyle F(s)= \int_0^\infty e^{-st}f(t) dt\)

With \(\displaystyle i(x)\) defined to \(\displaystyle 1.25(1- e^{-20t})\) for x from 0 to 1 and \(\displaystyle 1.25(e^{-20(t-1)}- e^{-20t})\) for x larger than 1, the Laplace transform is
\(\displaystyle 1.25\int_0^1 e^{-st}(1- e^{-20t})dt+ 1.25\int_1^\infty e^{-st}(e^{-20(t- 1)}- e^{-20t})dt\)
\(\displaystyle = 1.25\int_0^1 e^{-st}dt- 1.25\int_0^1 e^{(-s+ 20)t}dt+ 1.25e^{20}\int_1^\infty e^{(-s- 20)t}dt- \int_1^\infty e^{(-s- 20)t} dt\).

Surely you can do those integrals?