integrate f(x) from 0 to n. where n is a positive integer and f(x) is the greatest integer function.
I Know that i have to use the definiton of greatest integer function and break it into sum of the integrals how do i do this.
integrate f(x) from 0 to n. where n is a positive integer and f(x) is the greatest integer function.
I Know that i have to use the definiton of greatest integer function and break it into sum of the integrals how do i do this.
Okay.
A defined function on a closed interval is Riemann integrable if and only if it is continous almost everywhere.
We we need to find,
$\displaystyle \int_0^n [ x ] dx$
Since it is continous almost everywhere we can use the subdivision rule,
$\displaystyle \sum_{k=1}^n \int_{k-1}^k [x] dx$
But,
$\displaystyle \int_{k-1}^k [x] dx =k-1$
Because in that $\displaystyle [x]=k-1$ on this interval.
Thus,
$\displaystyle \sum_{k=1}^n k-1=\frac{n(n-1)}{2}$