Question

`int_((-pi)/8)^(pi/8) log ((2 - sin x)/(2 + sin x))` dx = ______.

Options

• `pi/8`

• `pi/2`

• `pi/4`

• 0

Answer 1

`int_((-pi)/8)^(pi/8) log ((2 - sin x)/(2 + sin x))` dx = 0.

Explanation:

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))`

= - f(x)

∴ f(x) is an odd function.

∴ `int_((-pi)/8)^(pi/8) log ((2 - sin x)/(2 + sin x))` dx = 0