Let $C$ be a positive definite matrix; the following multivariate integral is easy to calculate:
$\int \frac{d^n\vec{v}}{\sqrt{(2\pi)^{n}|\det C|}}e^{-\frac{1}{2}\vec{v}^TC^{-1}\vec{v}}\vec{v}^TC^{-1}\vec{v}=n$
However, in my work on statistical mechanics I would like to calculate the integral on the positive domain of $\vec{v}$, i.e. $0\le v_i\ \forall i=1..n$:
$\int_0^\infty \frac{d^n\vec{v}}{\sqrt{(2\pi)^{n}|\det C|}}e^{-\frac{1}{2}\vec{v}^TC^{-1}\vec{v}}\vec{v}^TC^{-1}\vec{v}=?\ (1)$
Alternatively, it may suffice for me to calculate a related integral:
$\int \frac{d^n\vec{v}}{\sqrt{(2\pi)^{n}|\det C|}}e^{-\frac{1}{2}\vec{v}^TC^{-1}\vec{v}}|\vec{v}|^TC^{-1}|\vec{v}|=?\ (2)$
where $|\vec{v}|_i=|v_i|$. Any directions or ideas?