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 $x$.

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