Question

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

Answer 1

$I=\int \frac{dx}{x^7(1+x^{-6})}$

Let $x^{-6}+1=t$

$\Rightarrow -6x^{-7}dx=dt$

$\Rightarrow I=\frac{-1}{6}\int \frac{dt}{t}$

$\Rightarrow I=\frac{-1}{6}ln|1+\frac{1}{x^6}|+c$.