Convergence of $\int_{\mathbb{R}^{n}}\frac{1}{1+|x|^{k}}dx$
Change to polar.
$$\int_{\mathbb{R}^n} {dx\over 1 + |x|^k} = c_n\int_0^\infty {r^{n-1}\, dr\over 1 + r^k},$$ where $c_n$ is just the surface area of $S^{n-1}$ in $\mathbb{R}^n.$
Convergence of $\int_{\mathbb{R}^{n}}\frac{1}{1+|x|^{k}}dx$
Change to polar.
$$\int_{\mathbb{R}^n} {dx\over 1 + |x|^k} = c_n\int_0^\infty {r^{n-1}\, dr\over 1 + r^k},$$ where $c_n$ is just the surface area of $S^{n-1}$ in $\mathbb{R}^n.$