The greatest integer function?

Question

We define a function [x] (Greatest Integer Function) as
f(x) = [x] =
{ n, n ≤ x < n + 1, n is an integer

I correctly define the antiderivative ∫[x]dx as
∫[x]dx =
{ nx - (n² + n)/2, n ≤ x < n + 1, n is an integer

Now, how can you explain the constant quantity -(n² + n)/2? Where does it come from?

^_^
thanks!^_^
...easy...



Note:

You can test the correctness of that antiderivative by picking examples.

E.g. you may want to find the area under the function from -3.5 to 4, etc. you may find the area using the geometric method and using the antiderivative, and see that they are the same...

^_^

Answer 1

Th -(n²+n)/2 comes from the fact that the sum of the first n numbers is (n²+n). For integral values of x, we have ∫[x] dx is
0+1+2...+(x-1), which is clearly [k=0, x-1]∑k, or (x²-x)/2. For nonintegral values of x, we have ∫[x] dx from 0 to n as ∫[x] dx from 0 to [n] + [n](n-[n]). Thus solving:
([n]²-n)/2 - [n]² + [n]n
[n]n-([n]²+n)/2

Which agrees with the above formula.

Answer 2

cut up this into the sum of quite a few integrals on the era being integrated. to illustrate, you recognize that when the ideal integer function = 0, then you quite're integrating the 0-function. while it equals a million, you're integrating 6x^2. while it equals 2, you're integrating 12x^2 and so on.