Evaluate: ∫₀^(π/4) log(1 + tan x) dx

Let $I = \int_0^{\pi/4} \log(1+\tan x)\,dx$. Using $\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx$: $I = \int_0^{\pi/4} \log\left(1+\tan\left(\frac{\pi}{4}-x\right)\right)dx = \int_0^{\pi/4}\log\frac{2}{1+\tan x}\,dx$ Adding: $2I = \int_0^{\pi/4}\log 2\,dx = \frac{\pi}{4}\log 2$ Hence $I = \frac{\pi}{8}\log 2$.