answer..ineteg

like a lot of other questions, i think there is a problem with the variable of integration and the limits of the integral..

but assuming that they mean the right thing,

between I and I+1/2, the number is 1 while between I+1/2 and I it is 0

so between I to I+1, the first bracket gives us 1/2

So the above integral becomes

\int_{0}^{[x]}{1/2[x]dx}=\sum_{r=0}^{[x]-1}{\int_{r}^{r+1}{r/2}}=\sum_{r=0}^{[x]-1}{r/2}=1/2\times[0+1+2+...+r-1]=\frac{[x].([x]-1)}{4}