value of the value expression F (x)=[1+sinx]+[2+sin (x/2)]+[3+sin (x/3)]+.........+[n+sin (x/n)] for x belongs to 0 to pi. Where []denotes greatest integer function.
I'd converted f(x) as
f (x)=1+2+3+4+5+........+n+sin x+sin (x/2)+sin (x/3)+........+sin (x/n)-fractional part (1+sinx)-fractional part (2+sinx/2)-fractional part (3+sinx)+..
How to solve the fractional part ?
  Ans is (n^2+n-2)/2, (n^2+n)/2, (n^2+n+2)/2

If x = $\frac{\pi}{2}$ , then $sin(x) = 1$
If x = $\pi$ , then sin($\frac{x}{2}) = 1 $
In these cases, one of the sine-terms is 1.
For all other x, $0\le sin(\frac{x}{n}) < 1$ for all n.
So the great-integer function is zero for all sine-terms except 
for the term $sin(\frac{\pi}{2})=1$, for which it is 1.
So, for x = $\frac{\pi}{2},$ we have the sum
2 + 2 + 3 + ... + n = $\frac{n(n+1)}{2}+1 = \frac{n^2+n+2}{2}$
For x = $\pi,$ we have the sum
1 + 3 + 3 + ... + n =$\frac{n(n+1)}{2}+1 = \frac{n^2+n+2}{2}$
and for all other x we have simply the sum
1 + 2 + 3 + ... + n = $\frac{n(n+1)}{2} = \frac{n^2+n}{2}$
