If `[x]` denotes the greatest integer less than or equal to `x ,` then find the value of the integral `int_0^2x^2[x]dxdot`

Explanation

To solve the integral \( \int_0^2 x^2 [x] \, dx \), where \([x]\) denotes the greatest integer less than or equal to \(x\), we need to break the integral into parts based on the behavior of the greatest integer function.

### Step 1: Identify the intervals for \([x]\)
The function \([x]\) changes its value at integer points. Therefore, we will split the integral at these points:
- From \(0\) to \(1\), \([x] = 0\)
- From \(1\) to \(2\), \([x] = 1\)

### Step 2: Split the integral
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