A) \[-\frac{4}{{{({{x}^{4}}+1)}^{1/4}}}+c\]
B) \[\frac{1}{{{({{x}^{4}}+1)}^{1/4}}}+c\]
C) \[\frac{x}{{{({{x}^{4}}+1)}^{1/4}}}+c\]
D) None of the above
Correct Answer: C
Solution :
Let \[I=\int{\frac{1}{{{({{x}^{4}}+1)}^{5/4}}}}dx\] \[=\int{\frac{1}{{{x}^{5}}{{\left( 1+\frac{1}{{{x}^{4}}} \right)}^{5/4}}}}dx\] Put \[1+\frac{1}{{{x}^{4}}}=t\] \[\Rightarrow \] \[-\frac{4}{{{x}^{5}}}dx=dt\] \[\therefore \] \[I=-\frac{1}{4}\int{\frac{dt}{{{t}^{5/4}}}}=-\frac{1}{4}.\frac{{{t}^{-1/4}}}{-\frac{1}{4}}+c\] \[=\frac{1}{{{\left( 1+\frac{1}{{{x}^{4}}} \right)}^{1/4}}}+c\] \[=\frac{x}{{{(1+{{x}^{4}})}^{1/4}}}+c\]
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