I need help with this multiple integral
$$ I = \int_{-\infty}^{\infty} x^{T}A x \phantom{\times} \textrm{exp} \left( x^{T}Bx-\ln{x^{T}x} \right) dx, $$ where $x\in\mathbb{R}^n$, $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times n}$.
Following assumptions can be made:
- $A$ and $B$ is symmetric, negative definite and invertible
- One parameter of $x$ is constant (let's say $x_n = -1$)
This is not the answer but here are some ideas: $$ I = \int_{\mathbb S^{n-1}} u^T A u \int_0^\infty r^{n-1} e^{-r^2\,u^T (-B) u} dr du $$ Change of variable $r \mapsto r/\sqrt{u^T(-B) u}$: $$ I = \int_{\mathbb S^{n-1}} \frac{u^T A u}{(u^T (-B) u)^{n/2}} \int_0^\infty r^{n-1} e^{-r^2} dr du $$ with $ \int_0^\infty r^{n-1} e^{-r^2} dr = \frac12 \Gamma(n/2) $. From B symmetric there is an orthogonal matrix $\Omega$ such that $B = \Omega D \Omega^T$ where D is diagonal. If $\Omega$ is a rotation, the spherical integral is invariant (change of variable $u \mapsto \Omega u$): $$ I = \frac12 \Gamma(n/2) \int_{\mathbb S^{n-1}} \frac{u^T \Omega^T A \Omega u}{(u^T D u)^{n/2}} du $$ Spectral decomposition of A might also help... Unless I messed up this integral should converge without the condition $x_n = -1$. Also, this problem looks like some weighted average of Rayleigh's quotient...