Almost Walli's Integrals

Calculus Level 5

$\large \displaystyle \int_0^\infty \sin (x^n) \text{ d}x$

For all positive rationals, $$n > 1$$, $$I_n$$ denotes the integral as above.

If $$P_n$$ denotes the product, $$\displaystyle \large \prod_{r=1}^{n-1} I_{\left( \frac{n}{r} \right) }$$, then evaluate the following limit $$L$$.

$L = \lim_{n \to \infty} \left( \sqrt{n} P_n \right)^{\frac{1}{n}}$

Enter the value of $$\displaystyle \lfloor 1000L \rfloor$$.

Details and Assumptions:

• $$\displaystyle \lfloor x \rfloor$$ denotes the greatest integer less than or equal to $$x$$.
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