Evaluate the following integral: ∫ log(2- sin x)/(2 + sin x)dx, x ∈ [-π/2, π/2]

Question

Evaluate the following integral:

\(\int\limits_{-\pi/2}^{\pi/2}log\cfrac{2-sin\text x}{2+sin\text x}d\text x \)

Answer 1

Let us assume 

I = \(\int\limits_{-\pi/2}^{\pi/2}log\cfrac{2-sin\text x}{2+sin\text x}d\text x \)....equation 1

By property, we know that

Thus

Thus

I = 0