$f ^{\prime}( x )=\left\{\begin{array}{ll}-\left(2-\sin \frac{1}{ x }\right)- x \left(-\cos \frac{1}{ x } \cdot\left(-\frac{1}{ x ^{2}}\right)\right) & x <0 \\ \left(2-\sin \frac{1}{ x }\right)+ x \left(-\cos \frac{1}{ x }\left(-\frac{1}{ x ^{2}}\right)\right) & x >0\end{array}\right.$
$f^{\prime}(x)=\left\{\begin{array}{l}-2+\sin \frac{1}{x}-\frac{1}{x} \cos \frac{1}{x} x<0 \\ 2-\sin \frac{1}{x}+\frac{1}{x} \cos \frac{1}{x} x>0\end{array}\right.$
$f^{\prime}(x)$ is an oscillating function which is non-monotonic in $(-\infty, 0) \cup(0, \infty)$.
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$f(x)=\left\{\begin{array}{cc}e^{\min \left[x^2, x-[x]\right\}}, & x \in[0,1) \\e^{\left[x-\log _e x\right]}, & x \in[1,2]\end{array}\right.$
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