The Integral of |x| exp(−ax²/2) over ℝ

In the setting of Lebesgue Measure on the Real Line: Standing Notation, let $\exp$ be the exponential function, let $a>0$ be a real number, and let $h_{a}:\mathbb{R}\to\mathbb{R}$ be the function $h_{a}(x)=|x|\exp(-ax^{2}/2)$.

(i) The integral. $h_{a}\in\mathcal{M}^{+}(\mathbb{R},\mathcal{B}(\mathbb{R}))$, and its integral is

$$\int_{\mathbb{R}}|x|\exp(-ax^{2}/2)\,d\lambda(x)=\frac{2}{a}.$$

(ii) Integrability. $h_{a}\in\mathcal{L}^{1}(\lambda)$.