Question

Integrate the following function with respect to x:
$\int \frac{1}{(x^2-1)}dx$

Answer 1

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

$\Rightarrow \int \frac{1}{2}\left[\frac{(x+1)-(x-1)}{(x-1)(x+1)}\right]dx$

$\Rightarrow \int \frac{1}{2}(\frac{1}{x-1}-\frac{1}{x+1})dx$

$\Rightarrow \frac{1}{2}\left(ln\right(x-1)-ln(x+1\left)\right)$

$=\frac{1}{2}ln\left(\frac{x-1}{x+1}\right)$