Quite happy as a sum, why the ratio?

Calculus Level 5

\[\displaystyle \int_{1}^{10} \dfrac{\{x\}}{\lfloor x \rfloor} \ dx\]

If the value of definite integral above can be written as \(\dfrac{a}{b}\) for coprime positive integers, find \(a+b\).

Details and assumptions:

  • Every \(x\in \mathbb{R}\) can be written as \(x=\lfloor x \rfloor + \{x\} \).

  • \(\lfloor x \rfloor\) denotes greatest integer less than or equal to \(x\).

  • \(\{x\} \) is the fractional part of \(x\).

Also see product and sum of squares.
Read the brilliant.org wiki for further details.

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