Question

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

Answer 1

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

$=\frac{1}{4}\int \frac{4x^{3}dx}{x^4(x^4-1)}$

Let $x^4=t,4x^{3}dx=dt$

$=\frac{1}{4}\int \frac{dt}{t(t-1)}$

$=\frac{1}{4}\int \frac{dt}{t-1}-\frac{1}{4}\int \frac{dt}{t}$

$=\frac{1}{4}\ln |t-1|-\frac{1}{4}\ln t+C$

$=\frac{1}{4}\ln \left(\frac{t-1}{t}\right)+C$

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