Let f(x)=In`[cos|x|+(1)/(2)]` where [.] denotes the greatest integer function, then `int_(x_(1))^(x_(2)) lim_(nto oo)(({f(x)}^(n))/(x^(2)+tan^(2)x))dx` is equal to, where `x_(1),x_(2) in [(-pi)/(6),(pi)/(6)]`

Options

A. 1
B. 2
C. -1
D. 0

Answer

D

Explanation

For any `x in [(-pi)/(6),(pi)/(6)]`
`(sqrt(3))/(2)le cos |x|le 1`
`rArr (sqrt(3)+1)/(2) lt cos|x|+(1)/(2) le +(1)/(2) rArr[cos|x|+(1)/(2)]=1`
Hence,`underset(x_(1))overset(x_(2))int underset(n to oo)lim(({f(x)}^(n))/(x^(2)+tan^(2)x))dx=underset(x_(1))overset(x_(2))int 0 dx =0`