The value of \(\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{dx}{1+\sqrt{\tan x}}\) is

\(\frac{\pi}{2}\) \(\frac{\pi}{6}\) \(\frac{\pi}{12}\) \(\pi\)

\(\frac{\pi}{12}\)

Use \(\int_{a}^{b}f\left(x\right) dx=\int_{a}^{b}\left(a+b-x\right)dx\)