Laplace tranform of $\frac{\sin(t)}{t}$

Question

I'm trying to find the Laplace transform of

$$\frac{\sin(t)}{t}$$

What I've tried: $\int_0^\infty e^{-st} \frac{\sin(t)}{t} dt$ = $\int_0^\infty e^{-st} \sin(t) dt \int_0^\infty e^{-st} ds$ but I don't think this is correct as I am getting undefined results. Thanks in advance for any help.

Answer 1

$L(\sin t) = \frac{1}{1+s^2}$. Hence $$L\left(\frac{\sin t}{t}\right) = \int_s^\infty \frac{du}{u^2+1} = \frac{\pi}{2} - \tan^{-1}{s} $$