[Integrales Indefinidas] Integrar, expresamos el numerador en la forma y operamos

Integrar $\int\dfrac{dx}{x(x^n+1)}$.

Solución:

Expresamos el numerador en la forma $1=(x^n+1)-x^n$ y operamos
$$\begin{align}\int\dfrac{dx}{x(x^n+1)}&=\int\dfrac{(x^n+1)-x^n}{x(x^n+1)}=\int\left[\dfrac{x^n+1}{x(x^n+1)}-\dfrac{x^n}{x(x^n+1)}\right]dx\\ &=\int\left[\dfrac{1}{x}-\dfrac{x^{n-1}}{x^n+1}\right]dx=\int\dfrac{1}{x}dx-\int\dfrac{x^{n-1}}{x^n+1}dx\\ &=\int\dfrac{1}{x}dx-\dfrac{1}{n}\int\dfrac{d(x^n+1)}{x^n+1}=\ln|x|-\dfrac{1}{n}\ln|x^n+1|+c\\ &=\ln|x|-\ln\sqrt[n]{|x^n+1|}+c=\ln\left(\dfrac{|x|}{\sqrt[n]{|x^n+1|}}\right)+c\end{align}$$
Por lo tanto
$$\int\dfrac{dx}{x(x^n+1)}=\ln\left(\dfrac{|x|}{\sqrt[n]{|x^n+1|}}\right)+c$$