Integral

[man111]

[tex]\displaystyle \int_{0}^{[x]}\left([x] - \left[x - \frac{1}{2}\right]\right)dx =[/tex]

where [tex][x] =[/tex] greatest Integer function

[Гост]

[tex]\exists n\in\mathbb{Z}:[x]=n-1, n-1\leq x < n,[/tex]
[tex]\int_0^{[x]}[t]-[t-\frac{1}{2}]dt=\sum_{k=1}^{n-1}\int_{k-1}^{k-\frac{1}{2}}k-1-(k-2)dt+\int_{k-\frac{1}{2}}^{k}k-1-(k-1)dt=\sum_{k=1}^{n-1}\frac{1}{2}=\frac{n-1}{2}=\frac{[x]}{2}[/tex]