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

**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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