Strange integral

Calculus Level 5

\DeclareMathOperatorsincsinc0k=0n\sinc(x2k+1)dx=π2 \DeclareMathOperator*{sinc}{sinc\,} \large \int_{0}^{\infty} \prod\limits_{k=0}^{n}\sinc\left(\frac{x}{2k+1}\right)dx=\, \frac{π}{2}

Find the largest value of nn which satisfies the above equation.

nn is a natural number.

\DeclareMathOperatorsincsinc\sinc(x)=sinxx. \DeclareMathOperator*{sinc}{sinc\,} \sinc (x)=\frac{\sin x}{x}.

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