A) \[\frac{1}{n}{{\log }_{e}}({{x}^{n}}+1)+c\]
B) \[\frac{1}{n}{{\log }_{e}}\left( \frac{1}{{{x}^{n}}+1} \right)+c\]
C) \[\frac{1}{n}{{\log }_{e}}\left( \frac{x}{{{x}^{n}}+1} \right)+c\]
D) \[\frac{1}{n+1}{{\log }_{e}}\left( \frac{{{x}^{n}}}{{{x}^{n}}+1} \right)+c\]
E) \[\frac{1}{n}{{\log }_{e}}\left( \frac{{{x}^{n}}}{{{x}^{n}}+1} \right)+c\]
Correct Answer: E
Solution :
Let \[I=\int{\frac{dx}{x({{x}^{n}}+1)}}\] \[\Rightarrow \] \[I=\int{\frac{{{x}^{n-1}}}{{{x}^{n}}({{x}^{n}}+1)}}dx\] Put \[{{x}^{n}}=t\] \[\Rightarrow \] \[n{{x}^{n-1}}dx=dt\] \[\Rightarrow \] \[{{x}^{n-1}}dx=\frac{1}{n}dt\] \[\therefore \] \[I=\frac{1}{n}\int{\left( \frac{1}{t}-\frac{1}{t+1} \right)}dt\] \[=\frac{1}{n}\log \left( \frac{t}{t+1} \right)=\frac{1}{n}\log \left( \frac{{{x}^{n}}}{{{x}^{n}}+1} \right)+c\]
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