JEE Math Practice Question

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 :

1. A.$$2 e-1$$
2. B.$$2 e-\frac{1}{2}$$
3. C.$$1+\frac{3 e}{2}$$
4. D.$$(e-1)\left(e^{2}+\frac{1}{2}\right)$$