Let $a_1,\dots,a_n, b$ be positive real numbers.

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