A) \[\frac{1}{n}\log \left( \frac{{{x}^{n}}}{{{x}^{n}}+1} \right)+c\]
B) \[\frac{1}{n}\log \left( \frac{{{x}^{n}}+1}{{{x}^{n}}} \right)+c\]
C) \[\log \left( \frac{{{x}^{n}}}{{{x}^{n}}+1} \right)+c\]
D) none of these
Correct Answer: A
Solution :
[a] \[I=\int{\frac{dx}{x({{x}^{n}}+1)}=\int{\frac{{{x}^{n-1}}}{{{x}^{n}}({{x}^{n}}+1)}dx}}\] Putting \[{{x}^{n}}=t\] so that \[n{{x}^{n-1}}dx=dt.\]i.e., We get \[{{x}^{n-1}}dx=\frac{1}{n}dt\] \[I=\int{\frac{\frac{1}{n}dt}{t(t+1)}}=\frac{1}{n}\int{\left( \frac{1}{t}-\frac{1}{t+1} \right)dt}\] \[=\frac{1}{n}(\log t-\log \,(t+1))+C\] \[=\frac{1}{n}\log \left( \frac{{{x}^{n}}}{{{x}^{n}}+1} \right)+C\]
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