# Laplace transform of $\int_1^\infty\frac{\cos t}{t}dt$

Is the result of the of Laplace transform of $\int_1^\infty\frac{\cos t}{t}dt$ equal to $\frac{\int_1^\infty\frac{\cos t}{t}dt}{s}$?

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Your integral is a constant, so... – David Mitra Jul 17 '12 at 1:51
ok, thanks. I just hope I don't have to solve the integral – hinafu Jul 17 '12 at 2:19
It does converge (it resembles a convergent alternating series) and is a particular value of the so called cosine integral. I'm not sure if its value can be explicitly computed. – David Mitra Jul 17 '12 at 2:25
You don't have to, unless you know the cosine integral. – J. M. Jul 17 '12 at 2:28
@David: "I'm not sure if its value can be explicitly computed." - there's no simpler/elementary closed form, but you can numerically evaluate it, of course. – J. M. Jul 17 '12 at 2:29

Yes, it is. Note that you have a definite integral which, indeed, converges (it is a variant of the Cosine Integral). As such, you are finding the Laplace transform of a constant function. Of course, for a function $f$ with rule $f(t)=a$, its Laplace transform is $F(s)={a\over s}$.