Show that $\int_{0}^{\infty}\frac{\cos(at)-\cos(bt)}{t} =\ln\frac{b}{a}$

Question

It should be using Laplace transform. I found similar problems already solved but I need this to be shown using Laplace transforms:

$$\int_{0}^{\infty}\frac{\cos(at)-\cos(bt)}{t} = \ln\frac{b}{a}$$

Answer 1

A related problem. Here is a start. You can follow the steps

1) consider the integral

$$ F(s)=\int_{0}^{\infty}\frac{\cos at-\cos(bt)}{t}e^{-st}dt$$

2) find $F'(s)$

3) calculate the resulting integral.

I think you can finish it now?