Let [ ] denote the greatest integer function. Then the value of the integral ₀¹ ( [ ⁻¹ x] + [ ⁻¹ x] ) d x is equal to :

Step-by-step solution

For x [0, 1] , the range of the inverse trigonometric functions are ⁻¹ x [0, /2] and ⁻¹ x [0, /2] . Since 1 For the first term, [ ⁻¹ x] = 0 when 0 ⁻¹ x For the second term, [ ⁻¹ x] = 0 when 0 ⁻¹ x Since 1 > /4 , we know that 1 In (0, 1) : [ ⁻¹ x] = 0 and [ ⁻¹ x] = 1 . The integrand is 0 + 1 = 1 . In ( 1, 1) : [ ⁻¹ x] = 0 and [ ⁻¹ x] = 0 . The integrand is 0 + 0 = 0 . In ( 1, 1) : [ ⁻¹ x] = 1 and [ ⁻¹ x] = 0 . The integrand is 1 + 0 = 1 . The integral I becomes: I = ₀^ 1 1 , d x + _ 1 ^ 1 0 , d x + _ 1 ¹ 1 , d x I =