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The value of int-2^2 (sin^2 x)/([x/pi]+1...

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

A

4 - sin 4

B

4

C

sin 4

D

0

Text Solution

Verified by Experts

The correct Answer is:
D
Let `I=int_(-2)^(2)(sin^(2)x)/(1/(2)+[(x)/(pi)]]dx`
Also , let `f(x)=(sin^(2)x)/(1/(2)+[(x)/(pi)]]`
Then , f `f(x)=(sin^(2)(-x))/(1/(2)+[-(x)/(pi)]]` (replacing x by - x)
`=(sin^(2)x)/(1/(2)+(-1-[(x)/(pi)])) [:'[-x]={{:(-[x]_(,) ,if,x!inI),(-1-[x]_(,),if,x!inI):}]`
`rArrf(-x)=-(sin^(2)x)/(1/(2)+[(x)/(pi)]]=-f(x)`
i.e. f(x) is odd function
`:. I=0[:'int_(-a)^(a)f(x)dx={0, if f 9x) " is odd function "2int_(0)^(a)f(x)dx, if f (x) " is even fubction"]`
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