The value of the definite integral ₀^ 3 1 1 + [x^2] , dx , where [ ] denotes the greatest integer function, is

Step-by-step solution

Let I = ₀^ 3 1 1 + [x^2] , dx . As x varies from 0 to 3 , x^2 varies from 0 to 3 . The integer breakpoints for x^2 in this interval are 1 and 2 . These correspond to x = 1 and x = 2 . We split the integral at these points: I = ₀¹ 1 1 + [x^2] , dx + ₁^ 2 1 1 + [x^2] , dx + _ 2 ^ 3 1 1 + [x^2] , dx In the interval x [0, 1) , x^2 [0, 1) [x^2] = 0 . In the interval x [1, 2 ) , x^2 [1, 2) [x^2] = 1 . In the interval x [ 2 , 3 ) , x^2 [2, 3) [x^2] = 2 . Substituting these values into the integrals: I = ₀¹ 1 1 + 0 , dx +