Question

Integrate the given equation $\frac{1}{x(x^4-1)}$

Answer 1

$\int \frac{1}{x(x^4-1)}dx$

$\Rightarrow \int \frac{1}{x^5-x}dx$

$\Rightarrow \int \frac{1}{x^5(1-\frac{1}{x^4})}dx$

Take $1-\frac{1}{x^4}=t\Rightarrow \frac{4}{x^5}d=dt$

$\Rightarrow \frac{1}{4}\int \frac{dt}{t}$

$\Rightarrow \frac{1}{4}\log t+C$

$\Rightarrow \frac{1}{4}\log (1-\frac{1}{x^4})+C$

$\Rightarrow \frac{1}{4}\log \left(\frac{x^4-1}{x^4}\right)+C$