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 :
Put \[{{x}^{n}}=t\Rightarrow n{{x}^{n-1}}dx=dt\] \[\Rightarrow \frac{n{{x}^{n}}}{x}\,dx=dt\Rightarrow \frac{1}{x}\,dx=\frac{dt}{nt},\] then it reduces to \[\int_{{}}^{{}}{\frac{dt}{nt(t+1)}}=\frac{1}{n}\left[ \int_{{}}^{{}}{\frac{dt}{t(t+1)}} \right]\] \[=\frac{1}{n}\left[ \int_{{}}^{{}}{\frac{1}{t}\,dt-\int_{{}}^{{}}{\frac{1}{t+1}\ dt}} \right]=\frac{1}{n}\log \frac{{{x}^{n}}}{{{x}^{n}}+1}+c\].
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