Question

Solve:
$\int \frac{1}{x(x^6+1)}dx$

Answer 1

$\int \frac{1}{x(x^6+1)}dx$

$\int \frac{1}{\left(\frac{1}{x^6+1}\right)x^7}dx$

Let $u=\frac{1}{x^6}+1$ $\Rightarrow dx=\frac{-x^7}{6}dx$

$\Rightarrow \frac{-1}{6}\int \frac{1}{6}au$

$\Rightarrow \frac{-1}{6}\left[ln\right[u\left]\right]$

$\Rightarrow \frac{-lnu}{6}$

$\Rightarrow \frac{-ln\left(\frac{1}{x^6+1}\right)}{6}+c$