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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