The value of `int_(-2)^(2)(sin^(2)x)/([(x)/(pi)]+(1)/(2))dx` where [.] denotes greatest integer function , is

Options

A. 0
B. sin 4
C. 4
D. 4- sin 4

Answer

A

Explanation

`int_(-2)^2(sin^2x)/([x/pi]+1/2)dx`
Let `f(x)=(sin^2x)/([x/pi]+1/2)`
`f(-x)=(sin^2(-x))/([-x/pi]+1/2)=(sin^2x)/(-1-[x/pi]+1/2) " " (because [x]+[-x]=-1, "when" x cancelin I)`
`=(sin^2x)/(-([x/pi]+1/2))=-f(x)` (f(x) is odd function )
`therefore int_(-2)^2 f(x)dx=0`