Question

Evaluate $\int \frac{dx}{x(x^n+1)}$
(where $C$ is constant of integration)

• A. $-\frac{1}{n}\ln |1+x^{-n}|+C$
• B. $\frac{1}{n}\ln |1+x^{-n}|+C$
• C. $-\frac{1}{n}\ln |1+x^n|+C$
• D. $\frac{1}{n}\ln |1+x^n|+C$

Answer 1

The correct option is A $-\frac{1}{n}\ln |1+x^{-n}|+C$

Let $I=\int \frac{dx}{x(x^n+1)}$
$\Rightarrow I=\int \frac{dx}{x\cdot x^n(1+x^{-n})}$
$\Rightarrow I=\int \frac{x^{-n-1}}{1+x^{-n}}dx$
Put $1+x^{-n}=t$
$-n{x}^{-n-1}dx=dt$

$\Rightarrow I=\int \frac{dt}{(-n)t}=\frac{-1}{n}\int \frac{dt}{t}=-\frac{1}{n}\ln |t|+C$

$\Rightarrow I=-\frac{1}{n}\ln |1+x^{-n}|+C$