 Sine integral

For real $x>0$, the sine integral approaches $\pi /2$. Exponential damping gives ${\int }_0^{\infty }{e}^{-at}\sin t/tdt=\arctan (1/a)$, so the limiting constant follows by letting $a\downarrow 0$. Integration by parts of the tail gives $\mathrm{Si}(x)=\pi /2-\cos x/x-\sin x/x^2+O(x^{-3})$.