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Evaluate the following integral:int(-pi/...

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

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Given:`int_(-pi/2)^(pi/2)log((2-sinx)/(2+sinx))dx`
`I=int_(-pi/2)^(pi/2)log((2-sinx)/(2+sinx))dx`
As we know that `int_a^b f(x)dx=int_a^b f(a+b-x)dx`
`I=int_(-pi/2)^(pi/2)log((2-sinx)/(2+sinx))dx`......(i)
`I=int_(-pi/2)^(pi/2)log((2-sin(-x))/(2+sin(-x)))dx` `I=int_(-pi/2)^(pi/2) log((2+sinx)/(2-sinx))dx`......(ii)
On adding (i) and (ii) we get
`2I= int_(-pi/2)^(pi/2)log 1dx=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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