# 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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