Question

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

Answer 1

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

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

Let $(1+\frac{1}{x^n})=t$

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

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

$=\frac{-1}{n}\int \frac{dt}{t}=\frac{-1}{n}log|t|+c$

$=\frac{-1}{n}log|1+\frac{1}{x^n}|+c$.