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Question
Evaluate the following:
`int_((-pi)/2)^(pi/2) log((2 + sin x)/(2 - sin x)) * dx`
Solution
Let I = `int_((-pi)/2)^(pi/2) log((2 + sin x)/(2 - sin x)) * dx`
Let f(x) = `log((2 + sin x)/(2 - sin x))`
∴ f(– x)= `log[(2 + sin (-x))/(2 - sin (-x))]`
= `log((2 - sin x)/(2 + sin x))`
= `-log((2 + sin x)/(2 + sin x))`
= – f(x)
∴ f is an odd function.
∴ `int_((-pi)/2)^(pi/2) f(x) * dx` = 0
∴ `int_((-pi)/2)^(pi/2)log((2 + sin x)/(2 - sin x)) * dx` = 0.
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