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 $x^n+1=t$ $n{x}^{n-1}dx=dt$ $=\frac{1}{n}\int \frac{dt}{t(t-1)}$ $=\frac{1}{n}\int \left(\frac{1}{t-1}-\frac{1}{t}\right)dt$ $=\frac{1}{n}\log \left(\frac{t-1}{t}\right)+c$ $=\frac{1}{n}\log \left(\frac{x^n}{x^n+1}\right)+c$