$$\int^{\infty}_{0}(\prod^{n}_{r=1}\cos\frac{x}{r})\frac{\sin(4x)}{x}dx=\frac{\pi}{2}$$ I highly suspect that the relation is true. How would I go about proving this? I thought it would be induction related but I cannot get anywhere with it as it quickly leads (through parts) to integrating $\frac{\sin x}{x}$
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4Does this answer your question? Explain why $\int_0^\infty\frac{\sin{4x}}{x}\prod\limits_{k=1}^n \cos\left(\frac{x}{k}\right) dx\approx\frac{\pi}{2}$ – According to that Q&A, your conjecture is true for $n \le 30$, but wrong for $n \ge 31$. – Martin R Apr 19 '20 at 09:34
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See also https://math.stackexchange.com/a/1022868/42969. – Both found with Approach0 – Martin R Apr 19 '20 at 09:37