Odd Powers

Notice that integrals of the form

$$\int_\infty^\infty x e^{-\alpha x^2} dx$$

can be done explicilty by change of variable (let u = $\alpha x^2$ etc.) so integrals of the form

$$\int_\infty^\infty x^{2n+1} e^{-\alpha x^2} dx$$

are also available by partial differentiation with respect to $\alpha$.