int (-pi/2)^(pi/2)log((2-sin x)/(2+sinx)...

` int _(-pi/2)^(pi/2)log((2-sin x)/(2+sinx))dx ` is equal to

A

1

B

3

C

2

D

0

Text Solution

Verified by Experts

The correct Answer is:

D

We have, `l=int_((-pi)/(2))^((pi)/(2))log((2-sin x)/(2+sin x))dx`
Let `f(x)=log((2-sin x)/(2+sin x))`
Then, `f(-x)=log((2-sin(-x))/(2+sin(-x)))`
`=log((2+sin x)/(2-sin x))=log((2-sin x)/(2+sin x))^(-1)`
`=-log((2-sin x)/(2+sin x))=-f(x)`
Then, f(x) is an odd function.
`therefore int_(-(pi)/(2))^((pi)/(2))f(x)dx=0`
[`because` If f(x) is an odd function, then `int_(-a)^(a)f(x)dx=0`]

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int _(-pi //2) ^(pi//2) log ((2-sin x)/(2+sin x))dx=

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`1`

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A

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B

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C

`log_(e)((2pi+3sqrt3)/(2(pi+3)))`

D

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