Question

$ \int{\frac{dx}{x({{x}^{n}}+1)}} $ is equal to:

• 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

Answer 1

Answer: (A)

$ \int{\frac{dx}{x({{x}^{n}}+1)}} $ Putting $ {{x}^{n}}+1=t $ $ n{{x}^{n-1}}dx=dt $ $ =\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 $