Three-dimensional gaussian integral with $(x + y + z)^2$ term

Question

I need to analytically perform the following integral:

$$\mathcal{I} = \int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty e^{\frac{-r^2}{2}}(x + y + z)^2 dx dy dz$$

I know the standard way to perform multi-dimensional gaussian integrals is to split them up and calculate each dimension seperately, but I do not know how to deal with $(x + y + z)^2$ term.

Answer 1

Let $X,\,Y,\,Z$ be three $N(0,\,1)$ IIDs so $X+Y+Z\sim N(0,\,3)$. Then$$\mathcal{I}=(2\pi)^{3/2}\Bbb E(X+Y+Z)^2=3(2\pi)^{3/2}.$$