Suppose
$$
f(y)=\int\_0^\infty e^{-yx}\frac{\sin{x}}{x}\,dx
$$
then show that $f(y)+\arctan(y)$ is constant and use this result to show that 
$$
\int\_0^\infty \frac{\sin{x}}{x}\,dx=\frac{\pi}{2}.
$$

MY THOUGHTS: So clearly we have to show that $f(y)+\arctan(y)=\frac{\pi}{2}$ for all $y$ so that 
$$
\frac{\pi}{2}=f(0)+\arctan(0)=\int\_0^\infty \frac{\sin{x}}{x}\,dx,
$$
but I have no idea how to begin to combine these two expressions to get the desired result. Any thoughts?