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

Options

A. 1
B. 2
C. `4- sin 4`
D. none of these

Answer

B

Explanation

We have,
` I=underset(-2)overset(2)int (sin^(2)x)/([(x)/(pi)]+(1)/(2))dx`
`rArr I=underset(-2)overset(0)int (sin^(2)x)/([(x)/(pi)]+(1)/(2))dx+underset(0)overset(2)int(sin^(2)x)/([(x)/(pi)]+(1)/(2))dx`
`rArr I=underset(-2)overset(0)int (sin^(2)x)/(-1+(1)/(2))dx+underset(0)overset(2)int(sin^(2)x)/(0+(1)/(2))dx`
`rArr I=-2underset(-2)overset(0)int sin^(2) x dx +2underset(0)overset(2)int sin^(2)x dx`
`rArr I=-2underset(-2)overset(0)int sin^(2) t dt +2underset(0)overset(2)int sin^(2)x dx, "where" t=-x`
`rArr I=-2underset(-2)overset(0)int sin^(2)x dx +2underset(0)overset(2)int sin^(2)x dx=0`