A) \[{{3}^{7/6}}-{{3}^{5/6}}\]
B) \[{{3}^{5/3}}-{{3}^{1/3}}\]
C) \[{{3}^{4/3}}-{{3}^{1/3}}\]
D) \[{{3}^{5/6}}-{{3}^{2/3}}\]
Correct Answer: A
Solution :
\[I=\int_{{}}^{{}}{\frac{1}{{{\cos }^{2/3}}x{{\sin }^{1/3}}x.\sin x}}dx\] \[=\int_{{}}^{{}}{\frac{{{\tan }^{2/3}}x}{{{\tan }^{2}}x}}.{{\sec }^{2}}x.dx\] \[=\int_{{}}^{{}}{\frac{{{\sec }^{2}}x}{{{\tan }^{4/3}}x}}.dx\]\[\{tan\,x=t,se{{c}^{2}}x\,dx=dt\}\] \[=\int_{{}}^{{}}{\frac{dt}{{{t}^{4/3}}}=\frac{{{t}^{-1/3}}}{-1/3}}=-3\left( {{t}^{-1/3}} \right)\]\[\Rightarrow I=-3\tan {{(x)}^{-1/3}}\] \[\left. \Rightarrow I=\frac{3}{{{(tan\,x)}^{1/3}}} \right|_{\pi /6}^{\pi /3}=-3\left( \frac{1}{{{\left( \sqrt{3} \right)}^{1/3}}}-{{\left( \sqrt{3} \right)}^{1/3}} \right)\] \[=3\left( {{3}^{1/3}}-\frac{1}{{{3}^{1/6}}} \right)={{3}^{7/6}}-{{3}^{5/6}}\]
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