Quite happy as a sum, why the ratio?

Calculus Level 5

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

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

Details and assumptions:

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

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

  • {x}\{x\} is the fractional part of xx.

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

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