[x] is a Gaussian function, which refers to the largest integer not greater than x, so we should consider the image of [x] 4. First, [x] 4 is a discontinuous function at an integer point (x=k, k is an integer), and at two adjacent integer points, the function value is constant (equal to the fourth power of the left integer).

We know that from the existence theorem of primitive function, if the function f(x) is continuous in a certain interval, then there is an primitive function whose derivative function f(x) is F(x). The function [x] 4 is continuous only when the interval [k, k+ 1) k is an integer, so it must be integrated by segments. The indefinite integral on [k, k+ 1] is (1/5) x 5+c, and k is an integer.