How does Laplace transform ℒ{ sin(t)/t } solves definite integral 0 to ∞ ∫ (sin(t)/t) dt

Question

How does the answer of the Laplace transform $$\mathcal L \left\{ \frac{\sin t}{t} \right\}= \frac{\pi}{2}-\tan^{-1}(s)$$ solve the definite integral

$$\int_0^{\infty} \frac{\sin t}{t} dt = \frac{\pi}{2} $$

How are they related? why does this solve the definite integral?

Thank you.

Answer 1

Your statement is

$$\int_0^{\infty} dt \frac{\sin{t}}{t} e^{-s t} = \frac{\pi}{2} - \arctan{s} $$

Plug in $s=0$ to both sides.

There are lots of ways to prove the LT. One way to do it is to use the FT relation for the sinc term.

Answer 2

You need to use Abel's theorem to take the limit as s goes to zero. Tis is needed because 0 is the boudary of the LT.