Evaluate `(int_(0)^(n)[x]dx)/(int_(0)^(n){x}dx)` (where `[x]` and `{x}` are integral and fractional parts of `x` respectively and `n epsilon N`).

Options

A. `(1)/(n-1)`
B. `(1)/(n)`
C. n
D. `n-1`

Answer

D

Explanation

`I_(1)=underset(0)overset(n)int{x}dx=underset(1)overset(2)int 0dx+underset(2)overset(3)int 2dx+...+ underset(n-1)overset(n)int(n-1)dx`
`rArr I_(2)=1(2-1)+2(3-2)+3(4-3)+...+ (n-1)(n-(n-1))`
`rArr =1+2+3+...+ (n-1)=(n(n-1))/(2)`
and,
`I_(2)=underset(0)overset(n)int{x}dx=underset(0)overset(n)int(x-[x])dx=(n)/(2)[ "Using graph of "(fx)=x-[x]]`
` :. (I_(1))/(I_(2))=n-1`