A) \[3{{\tan }^{-1/3}}x+C\]
B) \[-\frac{3}{4}{{\tan }^{-4/3}}x+C\]
C) \[-3{{\cot }^{-1/3}}x+C\]
D) \[-3{{\tan }^{-1/3}}x+C\]
Correct Answer: D
Solution :
\[I=\int_{{}}^{{}}{\frac{dx}{{{\left( \sin x \right)}^{4/3}}.{{\left( \cos x \right)}^{2/3}}}}\] \[I=\int_{{}}^{{}}{\frac{dx}{{{\left( \frac{\sin x}{\cos x} \right)}^{4/3}}.{{\cos }^{2}}x}}\] \[\Rightarrow I=\int_{{}}^{{}}{\frac{{{\sec }^{2}}x}{{{\left( \tan x \right)}^{4/3}}}}dx\] \[put\,\tan x=t\Rightarrow {{\sec }^{2}}xdx=dt\] \[\therefore \]\[I=\int_{{}}^{{}}{\frac{dt}{{{t}^{4/3}}}\Rightarrow I=\frac{-3}{{{t}^{1/3}}}}+c\] \[I=\frac{-3}{{{\left( \tan x \right)}^{1/3}}}+c\]
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