Definite Integration

Let the function \(f:[0,2] \rightarrow \mathbb{R}\) be defined as

$$f(x)= \begin{cases}e^{\min \left\{x^{2}, x-[x]\right\},} & x \in[0,1) \\ e^{\left[x-\log _{e} x\right]}, & x \in[1,2]\end{cases}$$

where \([t]\) denotes the greatest integer less than or equal to \(t\). Then the value of the integral \(\int_\limits{0}^{2} x f(x) d x\) is :