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The value of int(-pi//2)^(pi//2)(sin^(2)...

The value of `int_(-pi//2)^(pi//2)(sin^(2)x)/(1+2^(x))dx` is

A

`pi//4`

B

`pi//8`

C

`pi//2`

D

`4pi`

Text Solution

Verified by Experts

The correct Answer is:
A
`I=int_(-pi//2)^(pi//2) (sin^(2)x)/(1+2^(x))dx`…………….i
or `I=int_(-pi//2)^(pi//2) (sin^(2)(-x))/(1+2^(-x))dx`
or `I=int_(-pi//2)^(pi//2) (2^(x).sin^(2)x)/(2^(x)+1)dx`……………ii
Adding i and ii we get
`2I=int_(-pi//2)^(pi//2) sin^(2)xdx=2int_(0)^(pi//2) sin^(2)xdx`
`=int_(0)^(pi//2)(1-cos^(2)x)dx`
`=[x-(sin2x)/2]_(0)^(pi//2)=(pi)/2`
`:. I=(pi)/4`
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