Key Concepts and Formulas

• Piecewise Function Integration: To integrate a piecewise function, split the integral into sub-integrals corresponding to each piece of the function's definition.
• Greatest Integer Function ($[t]$): Understand how $[t]$ behaves over different intervals. For $x \in [0,1)$, $[x]=0$. For $x \in [1,2]$, we need to analyze $x-\log_e x$.
• Min Function: Determine $\min\{a, b\}$ by comparing $a$ and $b$. For $x \in [0,1)$, compare $x^2$ and $x$.
• Substitution Rule for Integration: If $u=g(x)$, then $\int f(g(x))g'(x)dx = \int f(u)du$. Remember to change integration limits when using substitution.
• Integration of Exponential Functions: $\int e^u du = e^u + C$.

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Step-by-Step Solution

Step 1: Define $f(x)$ for $x \in [0,1)$

The function is given by $f(x)=e^{\min \left\{x^{2}, x-[x]\right\}}$ for $x \in[0,1)$.

• Analyze $x-[x]$: For $x \in [0,1)$, the greatest integer $[x]$ is 0. Thus, $x-[x] = x-0 = x$.
• Analyze $\min\{x^2, x\}$: We compare $x^2$ and $x$ for $x \in [0,1)$. Consider $x^2 - x = x(x-1)$. For $x \in (0,1)$, $x>0$ and $x-1<0$, so $x(x-1)<0$, which means $x^2 < x$. At $x=0$, $x^2=0$ and $x=0$, so $x^2=x$. Therefore, for $x \in [0,1)$, we have $x^2 \le x$, which implies $\min\{x^2, x\} = x^2$.
• Result for $x \in [0,1)$: Substituting back, $f(x) = e^{x^2}$ for $x \in [0,1)$.

Step 2: Define $f(x)$ for $x \in [1,2]$

The function is given by $f(x)=e^{\left[x-\log _{e} x\right]}$ for $x \in[1,2]$.

• Analyze the exponent $g(x) = x - \log_e x$: To find $[g(x)]$, we analyze the behavior of $g(x)$ on $[1,2]$. Calculate the derivative: $g'(x) = \frac{d}{dx}(x - \log_e x) = 1 - \frac{1}{x}$.
• Determine Monotonicity: For $x \in [1,2]$, $x \ge 1$, so $0 < \frac{1}{x} \le 1$. Thus, $g'(x) = 1 - \frac{1}{x} \ge 0$. Specifically, $g'(1) = 1-1=0$, and for $x \in (1,2]$, $g'(x) > 0$. This means $g(x)$ is strictly increasing on $[1,2]$.
• Evaluate $g(x)$ at endpoints: $g(1) = 1 - \log_e 1 = 1 - 0 = 1$. $g(2) = 2 - \log_e 2$. Since $e \approx 2.718$, $\log_e 2 < \log_e e = 1$. More precisely, $\log_e 2 \approx 0.693$. So, $g(2) \approx 2 - 0.693 = 1.307$.
• Determine $[g(x)]$: The range of $g(x)$ on $[1,2]$ is $[g(1), g(2)] = [1, 2-\log_e 2]$. Since $1 \le g(x) \le 2-\log_e 2 \approx 1.307$, the greatest integer $[g(x)]$ is always 1 for $x \in [1,2]$.
• Result for $x \in [1,2]$: Substituting back, $f(x) = e^1 = e$ for $x \in [1,2]$.

Step 3: Rewrite the Piecewise Function $f(x)$

Combining the results from Step 1 and Step 2, the function $f(x)$ can be written as: $f(x)=\begin{cases}e^{x^2}, & x \in[0,1) \\ e, & x \in[1,2]\end{cases}$

Step 4: Evaluate the Definite Integral

We need to compute \int_\limits{0}^{2} x f(x) d x. We split the integral at $x=1$: \int_\limits{0}^{2} x f(x) d x = \int_\limits{0}^{1} x f(x) d x + \int_\limits{1}^{2} x f(x) d x Substitute the simplified forms of $f(x)$: \int_\limits{0}^{2} x f(x) d x = \int_\limits{0}^{1} x e^{x^2} d x + \int_\limits{1}^{2} x \cdot e \, d x

Step 4.1: Evaluate the first integral \int_\limits{0}^{1} x e^{x^2} d x

• Substitution: Let $u = x^2$. Then $du = 2x \, dx$, so $x \, dx = \frac{1}{2} du$. Change the limits: When $x=0$, $u=0^2=0$. When $x=1$, $u=1^2=1$.
• Integration: \int_\limits{0}^{1} x e^{x^2} d x = \int_\limits{0}^{1} e^u \left(\frac{1}{2} du\right) = \frac{1}{2} \int_\limits{0}^{1} e^u du $= \frac{1}{2} [e^u]_0^1 = \frac{1}{2} (e^1 - e^0) = \frac{1}{2} (e - 1)$

Step 4.2: Evaluate the second integral \int_\limits{1}^{2} x \cdot e \, d x

• Constant Factor: $e$ is a constant. \int_\limits{1}^{2} x \cdot e \, d x = e \int_\limits{1}^{2} x \, d x
• Integration: $= e \left[\frac{x^2}{2}\right]_1^2 = e \left(\frac{2^2}{2} - \frac{1^2}{2}\right)$ $= e \left(\frac{4}{2} - \frac{1}{2}\right) = e \left(2 - \frac{1}{2}\right) = e \left(\frac{3}{2}\right) = \frac{3e}{2}$

Step 5: Combine the Results

Add the results from Step 4.1 and Step 4.2: \int_\limits{0}^{2} x f(x) d x = \frac{1}{2}(e-1) + \frac{3e}{2} $= \frac{e}{2} - \frac{1}{2} + \frac{3e}{2}$ $= \frac{e+3e}{2} - \frac{1}{2}$ $= \frac{4e}{2} - \frac{1}{2}$ $= 2e - \frac{1}{2}$

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Common Mistakes & Tips

• Careful with $[x]$: Always determine the exact value of $[x]$ for the specific interval you are considering. For $x \in [0,1)$, $[x]=0$.
• Monotonicity Analysis: When dealing with $[g(x)]$, properly analyze the derivative of $g(x)$ to determine its increasing/decreasing behavior and find its range.
• Substitution Limits: Ensure that the limits of integration are correctly transformed when using the substitution rule.

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Summary

The problem requires evaluating a definite integral of a piecewise function. We first simplified the definition of $f(x)$ over the intervals $[0,1)$ and $[1,2]$ by carefully analyzing the greatest integer function and the minimum function. For $x \in [0,1)$, $f(x)=e^{x^2}$, and for $x \in [1,2]$, $f(x)=e$. The integral was then split into two parts, $\int_0^1 x e^{x^2} dx$ and $\int_1^2 xe dx$. The first integral was solved using a u-substitution ($u=x^2$), yielding $\frac{1}{2}(e-1)$. The second integral was a straightforward integration of a constant times $x$, resulting in $\frac{3e}{2}$. Combining these two results gives the final answer.

The final answer is $\boxed{2 e-\frac{1}{2}}$.