Integrate the rational function: $\frac{1}{x(x^{n}+1)}$

To integrate $\int \frac{1}{x(x^{n}+1)} dx$,we multiply the numerator and denominator by $x^{n-1}$:
$\int \frac{1}{x(x^{n}+1)} dx = \int \frac{x^{n-1}}{x^{n}(x^{n}+1)} dx$
Let $x^{n} = t$. Then $n x^{n-1} dx = dt$,which implies $x^{n-1} dx = \frac{dt}{n}$.
Substituting these into the integral:
$\int \frac{1}{n} \frac{dt}{t(t+1)} = \frac{1}{n} \int \left( \frac{1}{t} - \frac{1}{t+1} \right) dt$
Integrating the terms:
$= \frac{1}{n} [\log |t| - \log |t+1|] + C$
$= \frac{1}{n} \log \left| \frac{t}{t+1} \right| + C$
Substituting $t = x^{n}$ back:
$= \frac{1}{n} \log \left| \frac{x^{n}}{x^{n}+1} \right| + C$,where $C$ is an arbitrary constant.