TITLE: "sinc-ing" integral
QUESTION [7 upvotes]: Let $a_1,\dots,a_n, b$ be positive real numbers. 

*Question.** Is this true?
  $$\int_{-\infty}^{\infty}\frac{\sin(bx+a_1x+\cdots+a_nx)}{x}\prod_{j=1}^n\frac{\sin(a_jx)}{a_jx}\,\,dx=\pi.$$
  My most immediate quest is "why is it independent of $b$, in particular?"

REPLY [11 votes]: A more general result is due to C. Störmer (Acta Mathematica December 1895, Volume 19, Issue 1, pp 341–350)