The value of ₀^2 [x^2 ] d x is (where [x] denotes the greatest integer function not greater than x )

Step-by-step solution

To evaluate the integral ₀^2 [x^2 ] d x , split the integration interval where the greatest integer function [x^2 ] changes value. The critical points occur at x values where x^2 is an integer between 0 and 4 : x = 0 , 1 , 2 , 3 , and 2 . Break the integral into subintervals based on the behavior of [x^2 ] : ₀^2 [x^2 ] d x = ₀^1 [x^2 ] d x + ₁^ 2 [x^2 ] d x + _ 2 ^ 3 [x^2 ] d x + _ 3 ² [x^2 ] d x On [0,1) , 0 x^2 Substitute the constant values: ₀^2 [x^2 ] d x = ₀^1 0 d x + ₁^ 2 1 d x + _ 2 ^ 3 2 d x + _ 3 ² 3 d x E