I have a function $f:\mathbb R^n \to \mathbb R$ such that $f(x)=(1+|x|)^me^{-\frac{|x|^2}{a}}$.
I need to check is $$\int\limits_{\mathbb R^n}f(x)dx = \int\limits_{\mathbb R^n} (1+|x|)^me^{-\frac{|x|^2}{a}}dx$$
сonverges, but I have problem with finding integrable majorant. Could you please help me?
Here $m \geq 1$ is an arbitrary integer, $\alpha > 0$ is an arbitrary real.
Hint:
The expression $(1+|x|)^m$ is a polynomial in $|x|$. Thus for any positive constant $b$, there exists $K_b>0$ s.t. for all $|x|>K_b>0$, $e^{b|x|^2}>(1+|x|)^m$. Choose $b$ s.t. $(b-1/a)<0$. Now you can divide your integral in two parts: one for large $|x|$, where you have a integrable majorant, and one for small $|x|$.