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Can someone help me with this, - Integral Calculus - JEE Main-12

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}}

 
Answers (1)
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\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}}

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