Fractional parts in frenzy

Calculus Level 3

\[\frac {\displaystyle \int _{ 0 }^{ 12 }{ x\left\lfloor x \right\rfloor dx } }{\displaystyle \int _{ 0 }^{ 12 }{ \left\{ x \right\} x\left\lfloor x \right\rfloor dx } } =\frac { A }{ B } \]

If the above is true, where \(A\) and \(B\) are coprime positive integers, find \(\displaystyle A+B\)

Notation:

  • \(\left\{ x \right\} \) is the fractional part function.
  • \(\left\lfloor x \right\rfloor \) is the floor function.
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