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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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    @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.2012-07-17

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