Let the function $f:[0,2]\to\mathbb{R}$ be defined as \[ f(x)= \begin{cases} e^{\min\{x^2,\; x-[x]\}}, & x\in[0,1),\\[4pt] e^{[\,x-\log_e x\,]}, & x\in[1,2], \end{cases} \] where $[t]$ denotes the greatest integer less than or equal to $t$. Then the value of the integral $\displaystyle \int_{0}^{2} x f(x)\,dx$ is: