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Evaluate the following : int(-pi//2)^...

Evaluate the following :
`int_(-pi//2)^(pi//2)log((2-sinx)/(2+sinx))dx`

Text Solution

Verified by Experts

Let `I=int_(-pi//2)^(pi//2)log((2-sinx)/(2+sinx))dx`
If `f(x) =log ((2-sinx)/(2+sinx)),` then
`f(-x)=log[(2-sin(-x))/(2+sin(-x))]`
`=log((2+sinx)/(2-sinx))`
`=-log((2-sinx)/(2+sinx))=-f(x)`
`therefore ` f is an odd function.
`therefore `int_(-pi//2)^(pi//2)f(x)dx=0`
`"i.e. "int_(-pi//2)^(pi//2)log((2-sinx)/(2+sinx))dx=0.`
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Knowledge Check

  • int_(-pi/4)^(pi/4) log((2 - sinx)/(2 + sinx)) dx=

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    B
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    0

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