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 :
Let \[l=\int{\frac{dx}{x({{x}^{n}}+1)}=\int{\frac{{{x}^{n-1}}}{{{x}^{n}}({{x}^{n}}+1)}dx}}\] Put \[{{x}^{n}}+1=t\] \[\Rightarrow \] \[n{{x}^{n-1}}dx=dt\] \[\therefore \] \[l=\frac{1}{n}\int{\frac{dt}{t\,(t-1)}}\] \[=\frac{1}{n}\int{\left( \frac{1}{t-1}-\frac{1}{t} \right)dt}\] \[=\frac{1}{n}\log \left( \frac{t-1}{t} \right)+C\] \[=\frac{1}{n}\log \left( \frac{{{x}^{n}}}{{{x}^{n}}+1} \right)+C\]
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