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SOME USEFUL SHORTCUTS FOR FUNCTIONS :

1. $\int_{a}^{b}$ [ 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 $\in$ I

2. $\int_{a}^{b}$ [x]dx = n(n-1)/2 where a = 0 & b = n

where [.] denotes the greatest integer value function. n $\in$ I

3. $\int_{a}^{b}$ {x}dx = n/2 where a = 0 & b = n

where {.} denotes fractional function. n $\in$ I

4. $\int_{a}^{b}$ [x]dx = [n]*( (n+{n}-1)/2 ) where a = 0 & b = n

where [.] denotes the greatest integer value function. n $\in$ R

5. $\int_{a}^{b}$ {x}dx = n^2/2 - [n]( (n+{n}-1)/2 ) where a = 0 & b = n

where {.} denotes the fractional function. n $\in$ R

6. $\int_{a}^{b}$ {x}dx = (b-a)/2

where {.}denotes fractinol function. a,b $\in$ I

7. $\int_{a}^{b}$ [kx]dx = nk(nk-1)/2k where a = 0 & b= n

where [.] denotes greatest integer value function. n,k $\in$ I

8. $\int_{a}^{b}$ [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 $\in$ I

9. $\int_{a}^{b}$ [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 $\in$ I

10. $\int_{a}^{b}$ [sinkx]dx = - $\pi$ /k where a = 0 & b = 2 $\pi$ /k

where [.] denotes the greatest integer value function. k $\in$ I

11. $\int_{a}^{b}$ [coskx]dx = - $\pi$ /k where a = 0 & b = 2 $\pi$ /k

where [.] denotes greatest integer value function. k $\in$ I

12. $\int_{a}^{b}$ [x/k]dx = pn - (p(p+1)k)/2 where a = 0 & b= n

where p = [n/k], [.] denotes greatest integer function & n,k $\in$ I

13. $\int_{a}^{b}$ {kx}dx = n/2 where a = 0 & b = n

where {.} denotes fractional function. n $\in$ I

14. $\int_{a}^{b}$ {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 $\in$ I