Easy As A Sum, But The Sum Of Squares?

Calculus Level 2

\[\displaystyle \int_{0}^{12} \Bigl( \{x\}^2+\lfloor x \rfloor ^2\Bigr) \ dx = \ ? \]

Details and assumptions:

  • Every \(x\in \mathbb{R}\) can be written as \(x=\lfloor x \rfloor + \{x\} \).
  • As usual, \(\lfloor x \rfloor\) denotes greatest integer less than or equal to \(x\).
  • \(\{x\} \) is the fractional part of \(x\).
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