Question

Evaluate the following:

`int_((-pi)/2)^(pi/2) log((2 + sin x)/(2 - sin x)) * dx`

Answer 1

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.