integration - Compute $int_{0}^{infty}frac{sin(x)}{xe^x} dx$.

I know one can solve this in many ways and the answer is $\pi/4$. However I'm interested in one particular solution involving Laplace transform.

I once saw a solution of this integral where one just did something like directly taking the Laplace transform of the integrand and then getting an integral in terms of $s$. So my question is, how does one go from

$$\mathscr{L}\left[\int_{0}^{\infty}\frac{\sin{x}}{xe^x} \ dx\right]$$

To some integral like

$$\frac{1}{2}\int_0^{\infty}\frac{1}{s^2+1} \ ds = \frac{\pi}{4}.$$

I don't remember the steps inbetween or if the first step is even correct but I wan't to know how the Laplace theory was used here. Any one who has a guess?