SOME USEFUL SHORTCUTS FOR FUNCTIONS :
1. 
[ x ]dx = { (b-a)(b+a-1) }/2
= (upper limit - lower limit)(upper limit + lower limit - 1) / 2
where [.] denotes greatest integer value function a,b 
I
2. [x]dx = n(n-1)/2 where a = 0 & b = n
where [.] denotes the greatest integer value function. n I
3. {x}dx = n/2 where a = 0 & b = n
where {.} denotes fractional function. n I
4. [x]dx = [n]*( (n+{n}-1)/2 ) where a = 0 & b = n
where [.] denotes the greatest integer value function. n R
5. {x}dx = n^2/2 - [n]( (n+{n}-1)/2 ) where a = 0 & b = n
where {.} denotes the fractional function. n R
6. {x}dx = (b-a)/2
where {.}denotes fractinol function. a,b I
7. [kx]dx = nk(nk-1)/2k where a = 0 & b= n
where [.] denotes greatest integer value function. n,k I
8. [x^2] dx= -(1 + 2^1/2 + 3^1/2 + ...(n^2-1)^1/2) + (n^2-1)n where a= 0 & b= n
where [.] denotes the greatest integer value function. n I
9. [x^k]dx= -(1 + 2^1/k + 3^1/k +.... (n^k-1)^1/k )+ (n^k-1)n where a= 0 & b= n
where [.] denotes greatest integer value function. n,k I
10. [sinkx]dx = - 
/k where a = 0 & b = 2/k
where [.] denotes the greatest integer value function. k I
11. [coskx]dx = - /k where a = 0 & b = 2/k
where [.] denotes greatest integer value function. k I
12. [x/k]dx = pn - (p(p+1)k)/2 where a = 0 & b= n
where p = [n/k], [.] denotes greatest integer function & n,k I
13. {kx}dx = n/2 where a = 0 & b = n
where {.} denotes fractional function. n I
14. {x/k}dx = n^2/2k - pn + (p(p+1)k)/2 where a = 0 & b = n
where p = [n/k], {.} denotes fractional function & n,k I
