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

Question

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

Answer 1

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 $F(s)={a\over s}$.