Is there any exact formula or at least exact inequalities for the following intehral

$$
\int_{2}^{x}{{\rm d}t \over \left\lfloor\vphantom{\Large A}%
\log\left(x\right)/\log\left(t\right)\right\rfloor
\log\left(t\right)}
$$

where [x] is the greatest integer less than or equal to x.


added:

When I use 

$$
x-1<[x]\le x
$$ 
I get
$$
\frac{x-2}{\log x}=\int_2^x\frac{dt}{\log x}\leq \int_2^x\frac{dt}{[\frac{\log x}{\log t}]\log t}\le \int_2^x\frac{dt}{\log x-\log t}
$$
but they are not exact enough. I need more closer bounds.