Question

$I = \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}}\mspace{2mu}\frac{dx}{1 + \sqrt[3]{tan2x}}$

Apply king

$$I = \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}}\mspace{2mu}\frac{dx}{1 + \sqrt[3]{tan2\left( \frac{\pi}{4} - x \right)}}$$

$$\begin{array}{r} = \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}}\mspace{2mu}\mspace{2mu}\frac{dx}{1 + \sqrt[3]{cot2x}}\#(2) \end{array}$$

Add $(1) + (2) \Rightarrow 2I = \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}}\mspace{2mu}(1)dx$

$$I = \frac{1}{2}\left( \frac{\pi}{6} \right) = \frac{\pi}{12}$$