${\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}se{c}^{\frac{2}{3}}x\cos ec{c}^{\frac{4}{3}}xdx$ is 

The given integral is ${\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}se{c}^{\frac{2}{3}}x\cos ec{c}^{\frac{4}{3}}xdx$

The above integral can be written as 

${\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}se{c}^{\frac{2}{3}}x\cos ec{c}^{\frac{4}{3}}xdx={\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}\frac{se{c}^{2}x}{{\tan }^{{\displaystyle \frac{4}{3}}}x}dx$

Let $\tan x=t$, it implies that $se{c}^{2}xdx=dt$

The limits also $x=\frac{\pi }{6}\Rightarrow x=\frac{1}{\sqrt{3}}$ and $x=\frac{\pi }{3}\Rightarrow x=\sqrt{3}$

Hence, the integral is 

    $\begin{array}{rcl}{\int }_{\frac{1}{\sqrt{3}}}^{\sqrt{3}}\frac{1}{t^{{\displaystyle \frac{4}{3}}}}dt& =& {\left[-3t^{-\frac{1}{3}}\right]}_{\frac{1}{\sqrt{3}}}^{\sqrt{3}}\\ & =& \left(3^{\frac{7}{6}}-3^{\frac{5}{6}}\right)\end{array}$