I am reading Distribution Theory courently but there is part that I can't pass:
Last equation in $(1.204)$ makes no sence for me becouse as far as I know:
$\int_{-\infty}^{-\epsilon}\phi(x)ln|x|dx=-\int^{-\infty}_{-\epsilon}\phi(x)ln|x|dx=-\int^{-\infty}_{-\epsilon}\phi(x)ln(-x)dx=|t=-x,dx=-dt|=-\int^{\infty}_{\epsilon} \phi(-t)ln(t)(-1)dt=\int^{\infty}_{\epsilon}\phi(-t)ln(t)dt $
so where from is the minus in last formula?
You are completely right, it ought to be $[\varphi(x) + \varphi(-x)]$ in that formula, the "$-$" is a mistake. The conclusion that
$$\varphi \mapsto \int_\mathbb{R} \varphi(x)\ln \lvert x\rvert\,dx$$
is a distribution is unaffected, however.