The value of $\int_0^{\pi/3} \log(1 + \sqrt{3}\tan x)dx$ is equal to
Step-by-Step Solution
Key Concept: Algebraic simplification of nested logarithms and strategic use of substitution to resolve complex integrals
Let $I = \int_0^1 \log(1 + \sqrt{1+\tan x})dx$. Using the substitution and logarithm properties, we rewrite and simplify: $I = \int_0^1 \log\left(\frac{1+\sqrt{3+\tan x}}{1+\sqrt{1+\tan x}}\right)dx = \int_0^1 \log(4) - \log(1+\sqrt{1+\tan x})dx$. This gives $I = \frac{1}{2}\log 4 - \frac{1}{2}\log 2 = \frac{1}{2}\log 2$.
Correct Answer: 2