Partial fraction decomposition of even function I need to do partial fraction decomposition of this function (to solve its integral):
$\frac{t^2}{t^4+4}$
Since $t^4+4=(t^2+2t+2)(t^2-2t+2)$ I would do:
$\frac{t^2}{t^4+4}=A\frac{2t+2}{t^2+2t+2}+B\frac{1}{t^2+2t+2}+C\frac{2t-2}{t^2-2t+2}+D\frac{1}{t^2-2t+2}$
And then continue with calculations.
According to the teacher's notes though, I can instead write it like this because the function is even:
$\frac{t^2}{t^4+4}=A\frac{2t+2}{t^2+2t+2}+B\frac{1}{t^2+2t+2}-A\frac{2t-2}{t^2-2t+2}+B\frac{1}{t^2-2t+2}$
But I can't understand how to get it, knowing that the function is even.
Can you please help me?

Let $t=2u$
$$\dfrac{2u^2}{u^4+1}=\dfrac{u^2-1+u^2+1}{u^4+1}=\dfrac{1-1/u^2}{(u+1/u)^2-2}+\dfrac{1+/u^2}{(u-1/u)^2+2}=?$$
