Question

If [t] denotes the greatest integer \(≤ 1\), then the value of \(3\frac{(e - 1)^2}{e}\) is:

• A. e⁹ - e
• B. e⁸ - 1
• C. \(e^8 - e\)
• D. e⁹ - 1

Answer 1

The Correct Option is C

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).