Key Concepts and Formulas

1. Greatest Integer Function (GIF): $[t]$ denotes the greatest integer less than or equal to $t$. The value of $[t]$ is constant over intervals of the form $[n, n+1)$, where $n$ is an integer. This property is crucial for splitting definite integrals.
2. Properties of Definite Integrals:
   • Additivity over intervals: If $a < c < b$, then $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$. This is used to break down the integral based on the intervals where the GIF is constant.
   • Constant multiple rule: $\int_a^b c \cdot f(x) dx = c \cdot \int_a^b f(x) dx$.
3. Substitution Rule for Definite Integrals: If $u = g(x)$, then $du = g'(x) dx$. The limits of integration change from $x=a$ to $u=g(a)$ and from $x=b$ to $u=g(b)$.
4. Sum of a Geometric Progression (GP): The sum of the first $n$ terms of a GP with first term $a$ and common ratio $r$ is given by $S_n = a \frac{r^n - 1}{r-1}$.

Step-by-Step Solution

Let the given integral be $I$. $I = {{3(e - 1)} \over e}\int\limits_1^2 {{x^2}{e^{[x] + [{x^3}]}}dx}$

Step 1: Analyze the integrand and the limits of integration. The limits of integration are from $x=1$ to $x=2$. The integrand contains the terms $[x]$ and $[x^3]$. We need to determine how these terms behave over the interval $[1, 2]$.

Step 2: Evaluate the $[x]$ term within the interval $[1, 2]$. For $x \in [1, 2)$, $[x] = 1$. For $x = 2$, $[x] = 2$. However, since the integral is up to $2$, we consider the behavior just before $2$. For the purpose of splitting the integral, we will use the property that $[x]$ changes value at integer points.

Step 3: Evaluate the $[x^3]$ term within the interval $[1, 2]$. We need to find the integer values that $x^3$ takes as $x$ goes from $1$ to $2$. When $x=1$, $x^3 = 1$. So, $[x^3] = 1$. When $x$ increases from $1$, $x^3$ also increases. We need to find where $x^3$ becomes an integer greater than $1$. If $x^3 = 2$, then $x = \sqrt[3]{2} \approx 1.26$. If $x^3 = 3$, then $x = \sqrt[3]{3} \approx 1.44$. If $x^3 = 4$, then $x = \sqrt[3]{4} \approx 1.59$. If $x^3 = 5$, then $x = \sqrt[3]{5} \approx 1.71$. If $x^3 = 6$, then $x = \sqrt[3]{6} \approx 1.82$. If $x^3 = 7$, then $x = \sqrt[3]{7} \approx 1.91$. If $x^3 = 8$, then $x = 2$. So, $[x^3] = 8$ when $x=2$.

The integer values of $[x^3]$ in the interval $[1, 2]$ are $1, 2, 3, 4, 5, 6, 7$. These occur at $x$-values $1, \sqrt[3]{2}, \sqrt[3]{3}, \sqrt[3]{4}, \sqrt[3]{5}, \sqrt[3]{6}, \sqrt[3]{7}, 2$.

Step 4: Split the integral based on the values of $[x]$ and $[x^3]$. The integrand is $x^2 e^{[x] + [x^3]}$. For $x \in [1, \sqrt[3]{2})$, $[x] = 1$ and $[x^3] = 1$. So, $[x] + [x^3] = 1 + 1 = 2$. For $x \in [\sqrt[3]{2}, \sqrt[3]{3})$, $[x] = 1$ and $[x^3] = 2$. So, $[x] + [x^3] = 1 + 2 = 3$. For $x \in [\sqrt[3]{3}, \sqrt[3]{4})$, $[x] = 1$ and $[x^3] = 3$. So, $[x] + [x^3] = 1 + 3 = 4$. ... For $x \in [\sqrt[3]{7}, 2)$, $[x] = 1$ and $[x^3] = 7$. So, $[x] + [x^3] = 1 + 7 = 8$.

The integral can be split as follows: $\int\limits_1^2 {{x^2}{e^{[x] + [{x^3}]}}dx} = \int\limits_1^{\sqrt[3]{2}} {{x^2}{e^{1+1}}dx} + \int\limits_{\sqrt[3]{2}}^{\sqrt[3]{3}} {{x^2}{e^{1+2}}dx} + \int\limits_{\sqrt[3]{3}}^{\sqrt[3]{4}} {{x^2}{e^{1+3}}dx} + ... + \int\limits_{\sqrt[3]{7}}^{2} {{x^2}{e^{1+7}}dx}$ $= \int\limits_1^{\sqrt[3]{2}} {{x^2}{e^{2}}dx} + \int\limits_{\sqrt[3]{2}}^{\sqrt[3]{3}} {{x^2}{e^{3}}dx} + \int\limits_{\sqrt[3]{3}}^{\sqrt[3]{4}} {{x^2}{e^{4}}dx} + ... + \int\limits_{\sqrt[3]{7}}^{2} {{x^2}{e^{8}}dx}$

Step 5: Evaluate each integral. Each integral is of the form $\int_{a}^{b} c x^2 dx = c \left[ \frac{x^3}{3} \right]_a^b = \frac{c}{3} (b^3 - a^3)$.

The sum of the integrals is: $\frac{e^2}{3} ((\sqrt[3]{2})^3 - 1^3) + \frac{e^3}{3} ((\sqrt[3]{3})^3 - (\sqrt[3]{2})^3) + \frac{e^4}{3} ((\sqrt[3]{4})^3 - (\sqrt[3]{3})^3) + ... + \frac{e^8}{3} (2^3 - (\sqrt[3]{7})^3)$ $= \frac{e^2}{3} (2 - 1) + \frac{e^3}{3} (3 - 2) + \frac{e^4}{3} (4 - 3) + ... + \frac{e^8}{3} (8 - 7)$ $= \frac{e^2}{3} (1) + \frac{e^3}{3} (1) + \frac{e^4}{3} (1) + ... + \frac{e^8}{3} (1)$ $= \frac{1}{3} (e^2 + e^3 + e^4 + ... + e^8)$

Step 6: Recognize the sum as a Geometric Progression. The terms inside the parenthesis form a GP with first term $a = e^2$, common ratio $r = e$, and number of terms $n = 8 - 2 + 1 = 7$. The sum of this GP is $S_7 = e^2 \frac{e^7 - 1}{e - 1}$.

So, the value of the integral is: $\frac{1}{3} \left( e^2 \frac{e^7 - 1}{e - 1} \right) = \frac{e^2(e^7 - 1)}{3(e - 1)}$

Step 7: Multiply by the constant factor. The original expression is $I = {{3(e - 1)} \over e} \times (\text{the integral value})$. $I = {{3(e - 1)} \over e} \times \frac{e^2(e^7 - 1)}{3(e - 1)}$

Step 8: Simplify the expression. Cancel out the $3$ and $(e-1)$ terms. $I = \frac{1}{e} \times e^2 (e^7 - 1)$ $I = e (e^7 - 1)$ $I = e^8 - e$

Common Mistakes & Tips

• Incorrectly splitting the integral: Ensure that all points where $[x]$ or $[x^3]$ change their integer value within the integration interval are identified and used to split the integral. For $[x^3]$, this means finding $x$ such that $x^3$ is an integer.
• Forgetting to change limits of integration in substitution: If a substitution is used, remember to change the limits of integration accordingly. In this problem, we avoided direct substitution by evaluating the terms within each interval.
• Algebraic errors in GP sum: Double-check the first term, common ratio, and number of terms when applying the GP sum formula.

Summary

The problem involves evaluating a definite integral with the greatest integer function in the exponent. The key to solving this problem is to split the integral into sub-intervals based on the integer values taken by $[x]$ and $[x^3]$. Within each sub-interval, the exponent $[x] + [x^3]$ becomes a constant. After evaluating the integrals over these sub-intervals, the results form a geometric progression, which can be summed using the GP formula. Finally, the constant factor is applied to obtain the final answer.

The integral was split into intervals $[1, \sqrt[3]{2}), [\sqrt[3]{2}, \sqrt[3]{3}), \ldots, [\sqrt[3]{7}, 2)$. The exponent $[x] + [x^3]$ took values $2, 3, \ldots, 8$ over these respective intervals. The sum of the integrals resulted in $\frac{1}{3}(e^2 + e^3 + \ldots + e^8)$, which simplified to $\frac{e^2(e^7 - 1)}{3(e - 1)}$. Multiplying by the given constant $\frac{3(e - 1)}{e}$ yields the final answer $e^8 - e$.

The final answer is \boxed{e^8-e}.