Let the function f:[0,2] → R be defined as f(x)= {e^min {x^2, x-[x]}, x ∈[0,1); e^[x- log_e x], x ∈[1,2]} where [t] denotes the greatest integer less…

Let the function $\displaystyle f:[0,2] \rightarrow \mathbb{R}$ 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 $\displaystyle [t]$ denotes the greatest integer less than or equal to $\displaystyle t$. Then the value of the integral $\displaystyle \int_0^2 x f(x) d x$ is