The value of ₁^4 [x] d x where [x] is the greatest integer function less than or equal to x is equal to

Step-by-step solution

Evaluating the integral ₁^4 [x] dx requires handling the greatest integer function [x] by partitioning the domain where [x] remains constant—specifically at integer boundaries. Between x = 1 and x = 4 , the integral splits into three parts: ₁^4 [x] dx = ₁^2 1 dx + ₂^3 2 dx + ₃^4 3 dx . The first integral vanishes because 1 = 0 . The second evaluates to 2 (3-2) = 2 , while the third gives 3 (4-3) = 3 . Adding these yields 2 + 3 = (2 3) = 6 . Final answer: B