Question

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

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

Answer 1

Answer: (B)

$\frac{1}{n}\log \left(\frac{x^n+1}{x^n}\right)$

$\int \frac{dx}{x\left(x^n+1\right)}$

Expand

$\int \frac{1}{x^n}+1dx$

Substitute $u=\frac{1}{x^n}+1\to dx=-\frac{x^{n+1}}{n}du$

Now solving,

$=-\frac{1}{n}\int \frac{1}{u}du$

$=-\frac{1}{n}\log u=-\frac{1}{2}\log \left(\frac{1}{x^n}+1\right)=\frac{1}{n}\log \left(\frac{1+x^n}{x^n}\right)$