Question

Integrate the following functions w.r.t $x$:
$\frac{x}{x^4-2x^2-3}$

Answer 1

$I=\int \frac{x}{x^4-2x^2-3}dx=\int \frac{2xdx}{2(x^2-3)(x^2+1)}$

$x^2=t$

$I=\int \frac{dt}{2(t-3)(t+1)}=\frac{1}{8}\int \frac{(t+1)-(t-3)}{(t-3)(t+1)}dt$

$I=\frac{1}{8}\int \frac{1}{t-3}-\frac{1}{t+1}dt$

$=\frac{1}{8}ln\left(\frac{t-3}{t+1}\right)+c$