Why is this laplace identity true $\int_{\Bbb{R}^+}\frac{f(t)}{t}\,dt = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$?

Question

I was wondering why this laplace identity is true? Does it follow from definition? $$\int_{\Bbb{R}^+}\frac{f(t)}{t}\,dt = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$$

I'm trying to understand the first solution here how do you do this integral from fourier transform. I don't see where the identity comes from

Answer 1

In fact, you can use this in the proof: $$ \int_0^\infty e^{-st}dt=\frac{1}{s}, s>0. $$ So \begin{eqnarray} \int_0^\infty\frac{f(t)}{t}dt &=&\int_0^\infty f(t)\int_0^\infty e^{-st}ds\,dt\\ &=&\int_0^\infty\int_0^\infty e^{-st} f(t)\,dt\,ds\\ &=&\int_0^\infty L\{f(t)\}ds. \end{eqnarray} Done.