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