A) \[\log \left( \frac{{{x}^{7}}}{{{x}^{7}}+1} \right)+c\]
B) \[\frac{1}{7}\log \left( \frac{{{x}^{7}}}{{{x}^{7}}+1} \right)+c\]
C) \[\log \left( \frac{{{x}^{7}}+1}{{{x}^{7}}} \right)+c\]
D) \[\frac{1}{7}\log \left( \frac{{{x}^{7}}+1}{{{x}^{7}}} \right)+c\]
Correct Answer: B
Solution :
Given, \[\int_{{}}^{{}}{\frac{dx}{x\,({{x}^{7}}+1)}}=\int_{{}}^{{}}{\frac{dx}{{{x}^{8}}\left( 1+\frac{1}{{{x}^{7}}} \right)}}\] Put \[1+\frac{1}{{{x}^{7}}}=t\] Þ \[\frac{-7}{{{x}^{8}}}dx=dt\] \ \[I=\frac{-1}{7}\int{\frac{dt}{t}=}\frac{-1}{7}\log t+c\] Þ \[I=-\frac{1}{7}\log \left( \frac{{{x}^{7}}+1}{{{x}^{7}}} \right)+c\] Þ \[I=\frac{1}{7}\log \left( \frac{{{x}^{7}}}{{{x}^{7}}+1} \right)+c\].You need to login to perform this action.
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