`int_(0)^(2) [x^(2)]dx` is equal to

Options

A. `5-sqrt(2)`
B. `5-sqrt(3)-sqrt(2)`
C. `5-sqrt(3)`
D. none of these

Explanation

To solve the integral \( \int_{0}^{2} \lfloor x^2 \rfloor \, dx \), where \( \lfloor x^2 \rfloor \) is the greatest integer function, we will break down the integral into intervals where \( \lfloor x^2 \rfloor \) takes constant values.

### Step 1: Determine the intervals for \( \lfloor x^2 \rfloor \)

1. **Calculate the values of \( x^2 \) at the endpoints**:
- At \( x = 0 \), \( x^2 = 0 \)
- At \( x = 2 \), \( x^2 = 4 \)

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