# Gaussian multi-variate integral

I would like to compute the following integral $$I_n = \frac{1}{\sqrt{det(2\pi A)}} \int_{\mathbb{R}^n} ||x||^2_2 \exp\left(-\frac{1}{2} x^TAx\right) \mathrm{d} x$$ where $$A$$ is symmetric and positive definite.

Any suggestions or hints? Thanks in advance!

• Hint: by a change of variables, try to express it in terms of the variance of the standard Gaussian. – Federico Nov 6 '18 at 19:30

The strategy for dealing with this sort of integral is to start with the integral $$\int \exp\left(-\frac12 x^TAx+c^Tx\right)\,dx$$ for some fixed vector $$c$$.
If we then make the change of variables $$x\mapsto x+A^{-1}c$$, that integral becomes $$\int\exp\left(-\frac12 x^TAx+\frac12c^TA^{-1}c\right)\,dx=\sqrt{\det(2\pi A)}\exp\left(\frac12c^TA^{-1}c\right).$$
Now comes the trick: we differentiate the integral with respect to $$c_i$$ and $$c_j$$ (the $$i$$th and $$j$$th components of the $$c$$ vector), and then set $$c=0$$: $$\frac{\partial^2}{\partial c_i\partial c_j}\int \exp\left(-\frac12 x^TAx+c^Tx\right)\,dx=\sqrt{\det(2\pi A)}\frac{\partial^2}{\partial c_i\partial c_j}\exp\left(\frac12c^TA^{-1}c\right)$$ $$\int x_ix_j\exp\left(-\frac12 x^TAx\right)\,dx=\sqrt{\det(2\pi A)}(A^{-1})_{ij}$$ where $$(A^{-1})_{ij}$$ is the $$i,j$$ entry of the matrix $$A^{-1}$$.
Now you can just sum over the pairs $$x_ix_j$$ that you want; in your case you have $$\|x\|_2^2=\sum_{i=1}^nx_ix_i$$, so you get $$\sum_{i=1}^n (A^{-1})_{ii}=\boxed{\operatorname{Tr}A^{-1}}$$.