<s>[INST] partial fraction decomposition help now very quick way to solve 2x/x^2-4 as a partial fraction. I have tried the long way and it took over 30 minutes, I got it right but is it easier another way?
 [/INST] $$\frac{2x}{x^2-4}=\frac{2x}{(x-2)(x+2)}=\frac{A}{x-2}+\frac{B}{x+2}$$
Multiplying both sides by $x^2-4$ gives
$$2x=A(x+2)+B(x-2)=(A+B)x+2(A-B)$$
Hence $A+B=2$ and $A-B=0$. The second equation gives us $A=B$ and plugging this into the first equation gives us $A=B=1$.
Hence
$$\frac{2x}{x^2-4}=\frac{1}{x-2}+\frac{1}{x+2}$$
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