Find : ∫ log (1 + cosx) dx ; x ∈ (0,π)

Question

Find : \(\int\limits^{\pi}_0 \) log (1 + cosx) dx.

Answer 1

When f(2a – x) = f(x)

Adding (4) and (5), we get

Putting 2x = t

⇒ 2dx = dt ⇒ dx = 1/2 dt

When x = 0, then t = 0

When x = π/2 then t = π

From equation (6), we get

Putting the value of I₁ in equation (3), we get

I = 2I₁ = 2 (-π/2 log2) ⇒ I = -πlog 2