The value of the integral `int_(0)^(2) x[x] dx` is where [x] is greatest integer function.

Options

A. `7//2`
B. `3//2`
C. `5//2`
D. None of these

Answer

To solve the integral \( I = \int_{0}^{2} x [x] \, dx \), where \([x]\) is the greatest integer function, we first need to understand how the greatest integer function behaves over the interval from 0 to 2.

### Step 1: Break the integral at the points where the greatest integer function changes.
The greatest integer function \([x]\) changes at integer points. Therefore, we will break the integral into two parts:
1. From \(0\) to \(1\)
2. From \(1\) to \(2\)

Thus, we can write:
\[ I = \int_{0}^{1} x [x] \, dx + \int_{1}^{2} x [x] \, dx \]

### Step 2: Evaluate the first integral \(\int_{0}^{1} x [x] \, dx\).
In the interval \([0, 1)\), the greatest integer function \([x] = 0\). Therefore:
\[ \int_{0}^{1} x [x] \, dx = \int_{0}^{1} x \cdot 0 \, dx = \int_{0}^{1} 0 \, dx = 0 \]

### Step 3: Evaluate the second integral \(\int_{1}^{2} x [x] \, dx\).
In the interval \([1, 2)\), the greatest integer function \([x] = 1\). Therefore:
\[ \int_{1}^{2} x [x] \, dx = \int_{1}^{2} x \cdot 1 \, dx = \int_{1}^{2} x \, dx \]

### Step 4: Calculate \(\int_{1}^{2} x \, dx\).
The integral of \(x\) is given by:
\[ \int x \, dx = \frac{x^2}{2} \]
Now, we evaluate it from \(1\) to \(2\):
\[ \int_{1}^{2} x \, dx = \left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{2^2}{2} - \frac{1^2}{2} = \frac{4}{2} - \frac{1}{2} = 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2} \]

### Step 5: Combine the results.
Now we can combine the results from both integrals:
\[ I = 0 + \frac{3}{2} = \frac{3}{2} \]

### Final Answer:
Thus, the value of the integral \( I = \int_{0}^{2} x [x] \, dx \) is:
\[ \boxed{\frac{3}{2}} \]