how do I get the answer shown in the attached problem? when I tried it out, I couldn't get the numerator to 1.
Thanks!
We write
$\displaystyle \frac{2}{1-x^2}=\frac{2}{(1-x)(1+x)}=\frac{A}{1-x}+\frac{B}{1+x}_{.}$
With a bit of algebraic manipulation of the above equality(actually identity), we can find out that
$\displaystyle A = B = 1$.(There are several ways to do this. One of them is writing the right hand side as a single fraction and we get an equation by equating the denominator with 2)
So we have
$\displaystyle -1+\frac{2}{1-x^2} = -1 + \frac{2}{(1-x)(1+x)} = 1 +\frac{1}{1-x}+\frac{1}{1+x}_{.}$