Lebesgue Integral of slowly incrasing polynomial on $\mathbb{R}^m$

Question

I found that this fact is used everywhere in the study of Sobolov spaces and distribution theory: $\frac{1}{(1+|x|)^N}$ is integrable on $\mathbb{R}^m$ if $N>m$. But I am unable to find a proof for this.

Any kind of help is really appreciated.

Answer 1

Clearly the function is integrable on the unit ball $B(0,1)$. On the other hand $$\int_{\mathbf R^m \setminus B(0,1)} \frac{dx}{(1+|x|)^N} \le \int_{\mathbf R^m \setminus B(0,1)}|x|^{-N} \, dx = \int_1^\infty \int_{|x| = r} r^{-N}\, dS(x) dr = c \int_1^\infty r^{-N+m-1} \, dr$$ after a switch to spherical coordinates. The integral converges if $N > m$.