# The integral $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}sec^{\frac{2}{3}}x\: cosec^{\frac{4}{3}}x\: dx$ is equal to : Option 1) $3^{\frac{5}{6}}-3^{\frac{2}{3}}$ Option 2) $3^{\frac{4}{3}}-3^{\frac{1}{3}}$ Option 3) $3^{\frac{7}{6}}-3^{\frac{5}{6}}$ Option 4) $3^{\frac{5}{3}}-3^{\frac{1}{3}}$

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}sec^{\frac{2}{3}}x\: cosec^{\frac{4}{3}}x\: dx$

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{dx}{\frac{\sin^{\frac{4}{3}}x}{cos^{\frac{4}{3}}x}cos^{}\frac{2}{3}xcos^{}\frac{4}{3}x}$

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{sec^{2}x}{tan^{\frac{4}{3}}x}dx$

Let $tanx=t$

=> $sec^{2}xdx=dt$

$\int_{\frac{1}{\sqrt3}}^{\sqrt3}\frac{dt}{t^{\frac{4}{3}}}=-3[\frac{1}{t^{\frac{1}{3}}}]^{{\sqrt3}}_{\frac{1}{\sqrt3}}$

$=-3((\frac{1}{\sqrt3})^{\frac{1}{3}}-(\sqrt3)^{\frac{1}{3}})$

$=$  $3^{\frac{7}{6}}-3^{\frac{5}{6}}$

So, option (3) is correct.

Option 1)

$3^{\frac{5}{6}}-3^{\frac{2}{3}}$

Option 2)

$3^{\frac{4}{3}}-3^{\frac{1}{3}}$

Option 3)

$3^{\frac{7}{6}}-3^{\frac{5}{6}}$

Option 4)

$3^{\frac{5}{3}}-3^{\frac{1}{3}}$

Exams
Articles
Questions