A) \[\frac{4}{15}{{\left( 1-\frac{1}{{{x}^{3}}} \right)}^{5/4}}+c\]
B) \[\frac{4}{5}{{\left( 1-\frac{1}{{{x}^{3}}} \right)}^{5/4}}+c\]
C) \[\frac{4}{15}{{\left( 1+\frac{1}{{{x}^{3}}} \right)}^{5/4}}+c\]
D) None of these
Correct Answer: A
Solution :
\[\int_{{}}^{{}}{\frac{{{({{x}^{4}}-x)}^{1/4}}}{{{x}^{5}}}\,dx}=\int_{{}}^{{}}{\frac{1}{{{x}^{4}}}{{\left( 1-\frac{1}{{{x}^{3}}} \right)}^{1/4}}dx}\] \[=\frac{1}{3}\int_{{}}^{{}}{{{t}^{1/4}}dt}=\frac{4}{15}{{t}^{5/4}}+c=\frac{4}{15}{{\left( 1-\frac{1}{{{x}^{3}}} \right)}^{5/4}}+c\] \[\left\{ \text{Putting}\,1-\frac{1}{{{x}^{3}}}=t\,\text{and}\,\frac{1}{{{x}^{4}}}\,dx=\frac{1}{3}\,dt \right\}\]
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