Integral in $\mathbb{R}^n$ What the value of $$\int\limits_{|x|\geq1}\frac{1}{|x|^n}dx , \quad x \in \mathbb{R}^n?$$ This integral is part of other problem.

The integral
$$
K(n,\alpha) := \int\limits_{|x|\geq1}\frac{1}{|x|^\alpha}dx , \quad x \in \mathbb{R}^n?
$$
converges only if $\alpha > n$.  But as far as I know there is no known closed form.  Perhaps there are asymptotics as $\alpha \downarrow n$.
