It should be using Laplace transform. I found similar problems already solved but I need this to be shown using Laplace transforms:
$$\int_{0}^{\infty}\frac{\cos(at)-\cos(bt)}{t} = \ln\frac{b}{a}$$
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2Judging by the use of the word "need," this sounds like homework. If it is, please include whatever steps you have attempted so far. This helps any would-be answerers by not telling you something you already know (such dropping a hint you already figured out). – Clayton Oct 03 '13 at 13:21
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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?
Mhenni Benghorbal
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3Problem: $$\int_0^\infty \frac{\cos at}{t}e^{-st},dt$$ diverges due to the singularity in $0$. – Daniel Fischer Oct 05 '13 at 22:01
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@DanielFischer Could we fix this by replacing $\cos(at)$ with $\cos(at)-1$? I haven't checked the details, but perhaps differentiation under the integral sign works then? – Potato Oct 05 '13 at 22:04
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@DanielFischer And then the $1$s will cancel when we subtract, of course. – Potato Oct 05 '13 at 22:05
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1@Potato That would work. But why the detour, differentiating $$\int_0^\infty \frac{\cos at - \cos bt}{t}e^{-st},dt$$ directly would also work. – Daniel Fischer Oct 05 '13 at 22:07
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