Let [ ] and denote the greatest integer and fractional part functions, respectively. Then the value of the integral ₀⁹ [ x ] x d x is equal to :

Step-by-step solution

We know that the fractional part function is given by y = y - [y] . Therefore, the integral can be written as: I = ₀⁹ [ x ] ( x - [ x ] ) d x The function [ x ] changes its integer value when x is an integer, which occurs at x = 1, 4, 9 in the interval [0, 9] . We partition the integral into three sub-intervals: [0, 1] , [1, 4] , and [4, 9] . For x (0, 1) , [ x ] = 0 . For x (1, 4) , [ x ] = 1 . For x (4, 9) , [ x ] = 2 . Substituting these constant values into the integral: I = ₀¹ 0 ( x - 0) d x + ₁⁴ 1 ( x - 1) d