Question

Find: $\int \frac{1}{x(x^n+1)}dx$

Answer 1

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

Put $t=x^{-n}⟹dt=-n{x}^{-(n+1)}$

$⟹\int \frac{-dt}{n}\times \frac{1}{t+1}=-\frac{1}{n}\int \frac{dt}{t+1}=-\frac{1}{n}\ln |t+1|+C=-\frac{1}{n}\ln |x^{-n}+1|+C$