this seems like a homework problem. yes! To some extent. But really I was not getting it.
I was not able to get the Laplace transform of $\cos(t)/t$. using the property of Integration in Laplace Transform,
but since the integral is not convergent, I am not able to get the laplace transform.
but I know it exist!
then how to get the Laplace Transform?
how can It be possible that Using one Method you are getting and other (Integration method) you are not getting the answer ?