Obtain-a-formula-for-I-n-0-n-x-dx-in-terms-of-n-where-x-is-the-greatest-integer-function-of-x-

Question Number 126990 by physicstutes last updated on 26/Dec/20

Obtain a formula for

I_{n} = \underset{0}{\int^{n}} [x] d x in terms of n

where [x] is the greatest integer function of x .

Answered by mr W last updated on 26/Dec/20

I_{n} = \underset{0}{\int^{n}} [x] d x

= \underset{k = 0}{\sum^{n - 1}} \int_{k}^{k + 1} [x] d x

= \underset{k = 0}{\sum^{n - 1}} k \int_{k}^{k + 1} d x

= \underset{k = 0}{\sum^{n - 1}} k = \frac{n (n - 1)}{2}

Facebook Tweet Pin