If `[t]` denotes the greatest integer `let`,then the value of `(3(e-1))/(e)int_1^2x^(2)e^([x]+[x^3])dx` is :

Options

A. `e^(8)-1`
B. `e^(7)-1`
C. `e^(9)-e`
D. `e^(8)-e`

Answer

To solve the given problem, we need to evaluate the expression:

\[ \frac{3(e-1)}{e} \int_1^2 x^2 e^{[x] + [x^3]} \, dx \]

where \([x]\) denotes the greatest integer less than or equal to \(x\).

### Step 1: Determine the values of \([x]\) and \([x^3]\) in the integral.

For \(x\) in the interval \([1, 2]\):
- \([x] = 1\) since \(x\) is between 1 and 2.
- \([x^3] = [x^3]\) will vary depending on \(x\):
- When \(x = 1\), \(x^3 = 1\) and \([1] = 1\).
- When \(x = 2\), \(x^3 = 8\) and \([8] = 8\).

Thus, for \(x\) in \([1, 2)\), \([x^3] = 1\) when \(1 \leq x < 2\) and \([x^3] = 8\) when \(x = 2\). Therefore, we can consider the integral as:

\[ \int_1^2 x^2 e^{[x] + [x^3]} \, dx = \int_1^2 x^2 e^{1 + 1} \, dx = \int_1^2 x^2 e^2 \, dx \]

### Step 2: Simplify the integral.

Now we can simplify the integral:

\[ \int_1^2 x^2 e^2 \, dx = e^2 \int_1^2 x^2 \, dx \]

### Step 3: Calculate \(\int_1^2 x^2 \, dx\).

The integral \(\int x^2 \, dx\) can be computed as follows:

\[ \int x^2 \, dx = \frac{x^3}{3} + C \]

Evaluating from 1 to 2:

\[ \int_1^2 x^2 \, dx = \left[ \frac{x^3}{3} \right]_1^2 = \frac{2^3}{3} - \frac{1^3}{3} = \frac{8}{3} - \frac{1}{3} = \frac{7}{3} \]

### Step 4: Substitute back into the integral.

Now substituting back, we have:

\[ \int_1^2 x^2 e^2 \, dx = e^2 \cdot \frac{7}{3} \]

### Step 5: Substitute into the original expression.

Now substituting this result back into the original expression:

\[ \frac{3(e-1)}{e} \cdot e^2 \cdot \frac{7}{3} = \frac{(e-1) \cdot 7e^2}{e} \]

### Step 6: Simplify the expression.

This simplifies to:

\[ 7(e-1)e \]

### Step 7: Final result.

Thus, the final answer is:

\[ 7(e^2 - e) \]