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.