Question

For a real number x let `[x]` denote the largest integer less than or equal to x and `{x}=x-[x]`. The possible integer value of n for which `int_(1)^(n)[x]{x}dx` exceeds 2013 is
A. 63
B. 64
C. 90
D. 91

Answer 1

Correct Answer - D
`int_(1)^(n)[x]{x}dx=int_(1)^(n)[x](x-[x])dx`
`int_(1)^(2)(x-1)dx+int_(2)^(3)2(x-2)dx+int_(3)^(4)3(x-3)dx+int_(4)^(3)4(x-4)dx+…+int_(n-1)^(n)(n-1)(x-n+1)dx`
`=(1)/(2)+2(1)/(2)+3((1)/(2))+4((1)/(2))+…..+((n-1))/(2)`
`=(n(n-1))/(4)`
Check by options as to which will make it exceed 2013.