A remark concerning sinc integrals

We give a simple proof of Hanspeter Schmid’s result that $K_n:=2\int_0^\infty\cos t,\prod_{k=0}^n$\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}$\left(\frac{t}{2k+1}\right)$\mathrm{d}$t=\pi/2$if$n\in{0,1,\ldots,55}$,and$K_n<\pi/2$if$n\geq 56$. Furthermore, we present two sinc integrals where the value $\pi \mathrm{/}2$ is undercut as soon as $n\ge 418$ and $n\ge 3091$, respectively.