Fractional Part, Integrals, and Number Theory

Calculus Level 5

\[\int_{0}^{N} x\{x\} \,dx\]

\(X\) is the smallest positive integer \(N\) such that the integral above is an integer. If no such \(N\) exists, then let \(X=0.\)

\(Y\) is the smallest positive integer \(N\) such that the integral above is a perfect square. If no such \(N\) exists, then let \(Y=0.\)

What is \(X+Y\)?

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