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 that \[I=\int{\frac{d\,x}{x\,({{x}^{7}}+1)}}\] On putting \[{{x}^{7}}=t\] \[\Rightarrow \] \[7{{x}^{6}}dx=dt\] \[\Rightarrow \] \[dx=\frac{1}{7{{x}^{6}}}dt\] \[\therefore \] \[I=\int{\frac{1}{7{{x}^{7}}}\frac{dt}{(t+1)}}\] \[=\frac{1}{7}\int{\frac{1}{t}\frac{dt}{(t+1)}}\] \[(\because \,\,{{x}^{7}}=t)\] \[=\frac{1}{7}\int{\left( \frac{1}{t}-\frac{1}{t+1} \right)\,dt}\] \[=\frac{1}{7}\log \,\left( \frac{t}{(t+1)} \right)+c\] \[=\frac{1}{7}\log \,\left( \frac{{{x}^{7}}}{{{x}^{7}}+1} \right)+c\]
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