A) \[\frac{1}{5}\log {{x}^{5}}({{x}^{5}}+1)+c\]
B) \[\frac{1}{5}\log {{x}^{5}}\left( \frac{1+{{x}^{5}}}{{{x}^{5}}} \right)+c\]
C) \[\frac{1}{5}\log {{x}^{5}}\left( \frac{{{x}^{5}}}{{{x}^{5}}+1} \right)+c\]
D) None of these
Correct Answer: D
Solution :
We have \[I=\int{\frac{dx}{x({{x}^{5}}+1)}}=\int{\frac{dx}{{{x}^{6}}\left( 1+\frac{1}{{{x}^{5}}} \right)}}\] Put \[1+\frac{1}{{{x}^{5}}}=t\] Þ \[\frac{-5}{{{x}^{6}}}dx=dt\] Þ \[I=-\frac{1}{5}\int{\frac{dt}{t}=-\frac{1}{5}}\log t+c\] \[I=-\frac{1}{5}\log \left( 1+\frac{1}{{{x}^{5}}} \right)+c=-\frac{1}{5}\log \left( \frac{{{x}^{5}}+1}{{{x}^{5}}} \right)+c\] \ \[I=\frac{1}{5}\log \left( \frac{{{x}^{5}}}{{{x}^{5}}+1} \right)+c\].
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