Question

The value of \( \int \frac{dx}{x(x^{n} + 1)} \) is:

• A. $\frac{1}{n} \log \frac{x^{n}}{x^{n}+1} + c$
• B. $\log \frac{x^{n}}{x^{n}+1} + c$
• C. $\frac{1}{n} \log \frac{x^{n}+1}{x^{n}} + c$
• D. None of these

Hint:

Partial fractions are much easier to handle after a suitable substitution.

Answer 1

The Correct Option is A

Step 1: Concept
Multiply numerator and denominator by $x^{n-1}$.
Step 2: Analysis
$\int \frac{x^{n-1}}{x^{n}(x^{n}+1)} dx$. Let $x^{n} = t$, then $nx^{n-1} dx = dt$. $\frac{1}{n} \int \frac{dt}{t(t+1)} = \frac{1}{n} \int (\frac{1}{t} - \frac{1}{t+1}) dt$.
Step 3: Conclusion
$= \frac{1}{n} [\log t - \log(t+1)] = \frac{1}{n} \log \frac{t}{t+1} = \frac{1}{n} \log \frac{x^{n}}{x^{n}+1} + c$.
Final Answer: (A)