If [t] denotes the greatest integer \(≤ 1\), then the value of \(3\frac{(e - 1)^2}{e}\) is:
e9 - e
e8 - 1
\(e^8 - e\)
e9 - 1
The Correct Option is C
Solution and Explanation
Step 1: Substitution in the Integral
We are given the integral:
\[ \int_{1}^{2} x^2 e^{\lfloor x^3 \rfloor} dx. \]
Substitute \(t = x^3\), so \(3x^2 dx = dt\). The limits change as follows:
- When \(x = 1\), \(t = 1^3 = 1\).
- When \(x = 2\), \(t = 2^3 = 8\).
The integral becomes:
\[ \int_{1}^{2} x^2 e^{\lfloor x^3 \rfloor} dx = \frac{1}{3} \int_{1}^{8} e^{\lfloor t \rfloor} dt. \]
Step 2: Break the Integral Based on the Greatest Integer Function
Since the greatest integer function \(\lfloor t \rfloor\) takes integer values between \(1\) and \(8\), we can split the integral as:
\[ \int_{1}^{8} e^{\lfloor t \rfloor} dt = \int_{1}^{2} e^1 dt + \int_{2}^{3} e^2 dt + \cdots + \int_{7}^{8} e^7 dt. \]
Each integral evaluates to:
\[ \int_{k}^{k+1} e^k dt = e^k \cdot (k+1 - k) = e^k. \]
Thus, the summation becomes:
\[ \int_{1}^{8} e^{\lfloor t \rfloor} dt = e^1 + e^2 + e^3 + \cdots + e^7. \]
Step 3: Sum of Exponentials
The sum of exponentials is a geometric progression with first term \(1\), common ratio \(e\), and \(7\) terms:
\[ e^1 + e^2 + e^3 + \cdots + e^7 = e \cdot \left(1 + e + e^2 + \cdots + e^6\right). \]
The sum of the geometric progression is:
\[ 1 + e + e^2 + \cdots + e^6 = \frac{e^7 - 1}{e - 1}. \]
Thus:
\[ \int_{1}^{8} e^{\lfloor t \rfloor} dt = e \cdot \frac{e^7 - 1}{e - 1}. \]
Step 4: Multiply by the Given Coefficient
Now substitute back into the given expression:
\[ \frac{1}{3} \int_{1}^{8} e^{\lfloor t \rfloor} dt = \frac{1}{3} \cdot e \cdot \frac{e^7 - 1}{e - 1}. \]
Simplify:
\[ \frac{1}{3} \cdot e \cdot \frac{e^7 - 1}{e - 1} = \frac{e (e^7 - 1)}{3(e - 1)}. \]
Expand the denominator:
\[ \frac{e (e^7 - 1)}{3(e - 1)} = \frac{(e - 1)(e^7 - 1)}{3(e - 1)} = \frac{e^8 - e}{3}. \]
Conclusion
The value of the given expression is:
\[ \boxed{e^8 - e}. \]
Therefore, the correct answer is (3).
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