$|_{x=1}$ notation? I had the equation:
$$\frac{1}{(1+x)(1-x)^2}=\frac{A}{(1-x)^2}+\frac{B}{1-x}+\frac{C}{1+x}$$
$$A=\left.\frac{1}{1+x}\right|_{x=1} = \frac{1}{2} \tag2$$
What does $(2)$ mean? Particularly, the notation $|_{x=1}$

It means you substitute $x=1$ into the expression. Thus $$A=\left.\frac{1}{1+x}\right|_{x=1} =\frac{1}{1+1} = \frac{1}{2}. $$
This notation can also be used as follows:
$$\int_{0}^{1}\frac{1}{x+1}dx=\ln(x+1)\bigg|_{x=0}^{x=1}=\ln(1+1)-\ln(0+1)=\ln(2).$$
