Question

Evaluate $\int \frac{dx}{x\left(x^5+1\right)}$

Answer 1

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

$\Rightarrow \int \frac{dx}{x^6(1+x^{-5})}$

$1+x^{-5}=t$

$\frac{-5}{x^6}dx=dt$

$\Rightarrow \int -\frac{dt}{5\left(t\right)}$

$=\frac{-1}{5}ln\left|t\right|+c$

$=\frac{-1}{5}ln|1+x^{-5}|+c$