This post is inspired in the Borwein integral, and in a problem proposed by Ovidiu Furdui in Crux Mathematicorum, that is the Problem 3707, in page 151.

I've considered integrals of the form $$\int_0^\infty\prod_{k=1}^n\frac{\sin^2\left(\frac{x}{k}\right)}{\pi^2-\left(\frac{x}{k}\right)^2}dx\tag{1}$$ for integers $n\geq 1$. My belief is that $(1)$ isn't exactly Ovidiu's integral.

Question.Can you show that the case $n=3$ the integral in $(1)$ is $\neq 0$? Is it possible to find $n>2$ such that the integral in $(1)$ is equals to $0$ again?Many thanks.

Please if the integral in $(1)$ is Ovidiu's integral, or it is in the literatute, add a comment. I think that isn't the same integral, using Wolfram Alpha online calculator an alternate form of the difference of integrands, for the second case is showed as using the input

`sin^2(x)/(pi^2-x^2)sin^2(x/2)/(pi^2-(x/2)^2)- sin^4(x)/((pi^2-x^2)(4pi^2-x^2))`

## References:

[1] *Borwein integral*, from the encyclopedia Wikipedia.

[2] *Problem 3707*, Crux Mathematicorum, Volume 38, Number 4, April 2012.