Easy As A Sum, But The Sum Of Squares?

Calculus Level 2

012({x}2+x2) dx= ?\displaystyle \int_{0}^{12} \Bigl( \{x\}^2+\lfloor x \rfloor ^2\Bigr) \ dx = \ ?

Details and assumptions:

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