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

`int_(0)^(2)x^(2)[x]dx=int_(0)^(1)x^(2)[x]dx+int_(1)^(2)x^(2)[x]dx`
`=int_(0)^(1)x^(2)(0)dx+int_(1)^(2)x^(2)(1)dx`
`=0+int_(1)^(2)x^(2)dx=|(x^(3))/3|_(1)^(2)`
`=(8-1)/3=7/3`