A) \[\frac{1}{4}{{\left( x+\frac{1}{x} \right)}^{4}}+c\]
B) \[\frac{{{x}^{4}}}{4}+\frac{3{{x}^{2}}}{2}+3\log x-\frac{1}{2{{x}^{2}}}+c\]
C) \[\frac{{{x}^{4}}}{4}+\frac{3{{x}^{2}}}{2}+3\log x+\frac{1}{{{x}^{2}}}+c\]
D) None of these
Correct Answer: B
Solution :
\[\int_{{}}^{{}}{{{\left( x+\frac{1}{x} \right)}^{3}}dx=\int_{{}}^{{}}{\left( {{x}^{3}}+\frac{1}{{{x}^{3}}}+3x+\frac{3}{x} \right)\,dx}}\] \[=\frac{{{x}^{4}}}{4}-\frac{1}{2{{x}^{2}}}+\frac{3{{x}^{2}}}{2}+3\log x+c\] \[=\frac{{{x}^{4}}}{4}+\frac{3{{x}^{2}}}{2}+3\log x-\frac{1}{2{{x}^{2}}}+c.\]
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