# Elegance Comes Only Periodically

Calculus Level 5

$P\left( x \right) =\int _{ { -x }^{ 2 } }^{ 0 }{ \left( \prod _{ n=1 }^{ \infty }{ \frac { { n }^{ 2 }+t }{ { n }^{ 2 } } } \right) dt }$

When $$x \in \mathbb{Z}$$ and $$P(x) \neq 0$$ , the value of $$P\left( x \right)$$ can be expressed in the form $$\frac { \alpha }{ { \pi }^{ \beta } }$$, where $${ \alpha }$$ is a perfect square and $${ \beta }$$ is a prime number. Find $${ \alpha }^{ \beta }$$.

This problem is original. The picture of the graph was produced by Wolfram.

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