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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