What is the proof for the Laplace transform of f(t)/t?

[fishingspree2]

Hello, I am trying to find the laplace transform of $$\frac{f(t)}{t}$$

http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-KANPUR/mathematics-2/node92.html
Lemma 10.3.9

I don't understand why we are taking the limit as s goes to infinity

Can anyone help me?
thank you

 

[LCKurtz]

Look at it this way. You have this relationship:

$$F(s) = -\frac d {ds} G(s) = -G'(s)$$

Take the anitiderivative of both sides from s to b

$$\int_s^b F(u)\, du = -(G(b) - G(s)) = G(s)-G(b)$$

This must hold for all b, so let $b\rightarrow \infty$, knowing $G(b)\rightarrow 0$ and switching sides:

$$G(s) = \int_s^\infty F(u)\, du = \int_s^\infty F(s)\, ds$$

This builds in the constant of integration in the formula.