The integral \(\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sec ^{\frac{2}{3}} x \operatorname{cosec}^{\frac{4}{3}} x \mathrm{~d} x\) is equal to

Let \(\mathrm{I}=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sec ^{\frac{2}{3}} x \operatorname{cosec}^{\frac{4}{3}} x \mathrm{~d} x=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\mathrm{d} x}{\cos ^{\frac{2}{3}} x \cdot \sin ^{\frac{4}{3}} x}\)

\(=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sec ^2 x}{\tan ^{\frac{4}{3}} x} \mathrm{~d} x\)
Put \(\tan x=\mathrm{t} \Rightarrow \sec ^2 x \mathrm{~d} x=\mathrm{dt}\)
\(\therefore \quad I=\int_{\frac{1}{\sqrt{3}}}^{t^{\frac{\sqrt{3}}{3}}} \frac{\mathrm{dt}}{\frac{4}{3}}=\left[-3 \mathrm{t}^{-\frac{1}{3}}\right]_{\frac{1}{\sqrt{3}}}^{\sqrt{3}}\)
\(=-3\left[(\sqrt{3})^{\frac{-1}{3}}-\left(\frac{1}{\sqrt{3}}\right)^{\frac{-1}{3}}\right]\)
\(=-3\left(3^{-\frac{1}{6}}-3^{\frac{1}{6}}\right)\)
\(=3^{\frac{7}{6}}-3^{\frac{5}{6}}\)