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

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.

• Use Laplace of sint , then use.. division of t rule, which lets you integrate it from s to infty Nov 28, 2016 at 5:36
• I know how to do. I just don't know why it works?? Nov 28, 2016 at 5:56
• Go through the derivation of division of t rule. You will understand why it works Nov 28, 2016 at 8:05
• I understand the division of t rule and how it works. What i do not understand is by taking the Laplace transform ℒ{ sin(t)/t } it basically solve the definite integral? Nov 28, 2016 at 8:43

$$\int_0^{\infty} dt \frac{\sin{t}}{t} e^{-s t} = \frac{\pi}{2} - \arctan{s}$$
Plug in $s=0$ to both sides.
• $s=0$ is a boundary point, hence you need an argument when taking the limit. Dec 2, 2016 at 17:34