Evaluate the following integral: ∫ log(1 - cos x) dx, x ∈ [0, π]

Question

Evaluate the following integral: 

\(\int\limits_{0}^{\pi} \) log(1 - cos x) dx

Answer 1

Let us assume

I = \(\int\limits_{0}^{\pi} \) log(1 - cos x) dx

since log m n = log m + log n and log(m)ⁿ = n log m

since log(m/n) = log m - log n

Thus substituting the value of I₂ in equation 3

Substituting in equation (a) i.e