A) \[-{{\left( {{x}^{4}}+1 \right)}^{\frac{1}{4}}}+c\]
B) \[-{{\left( \frac{{{x}^{4}}+1}{{{x}^{4}}} \right)}^{\frac{1}{4}}}+c\]
C) \[{{\left( \frac{{{x}^{4}}+1}{{{x}^{4}}} \right)}^{\frac{1}{4}}}+c\]
D) \[{{\left( {{x}^{4}}+1 \right)}^{\frac{1}{4}}}+c\]
Correct Answer: B
Solution :
\[I=\int_{{}}^{{}}{\frac{dx}{{{x}^{5}}{{\left( 1+\frac{1}{{{x}^{4}}} \right)}^{{}^{3}/{}_{4}}}}}\] Let \[1+\frac{1}{{{x}^{4}}}={{t}^{4}}\] \[\frac{-4}{{{x}^{5}}}dx=4{{t}^{3}}dt\] \[I=\int_{{}}^{{}}{-\frac{{{t}^{3}}dt}{{{t}^{3}}}}=-t+c\] \[=-{{\left( \frac{{{x}^{4}}+1}{{{x}^{4}}} \right)}^{{}^{1}/{}_{4}}}+c\]
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