Question

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

Answer 1

Given,

$\begin{array}{c}\int \frac{1}{x\left(x^n+1\right)}dx\\ \int \frac{x^{n-1}}{x^n\left(x^n+1\right)}dx\end{array}$

Put $x^n+1=t\&n{x}^{n-1}dx=dt$

Therefore,

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

$=\frac{1}{n}[\int \frac{dt}{t-1}-\int \frac{dt}{t}]$

$=\frac{1}{n}\left\{\log \left(t-1\right)-\log t\right\}+C=\frac{1}{n}\left\{\log \left(1-\frac{1}{t}\right)\right\}+C=\frac{1}{n}\log \left(1-\frac{1}{x^n+1}\right)+C$

Hence, this is the answer.