Sine integral
= Sine integral
{title2=$\operatorname{Si}(x)=\int_0^x\sin t/t\,dt$}
For real $x>0$, the sine integral approaches $\pi/2$. Exponential damping gives $\int_0^\infty e^{-at}\sin t/t\,dt=\arctan(1/a)$, so the limiting constant follows by letting $a\downarrow0$. <Integration by parts> of the tail gives $\operatorname{Si}(x)=\pi/2-\cos x/x-\sin x/x^2+O(x^{-3})$.