I'm a student. In a recent assignment I was asked to find the mean of a Student's t multivariate distribution (which should be $\overline\mu$).
I've divided the integral required to find the expected value into two pieces, and one of the two,
$$\int_{-\infty}^{+\infty}\overline{z}\frac{1}{[1+\dfrac{1}{v}\overline{z}^T\mathbf\Sigma^{-1}\overline{z}]^\frac{v+p}{2}}d\overline{z}{}{}$$
is supposed to be zero. However since this is an improper integral I can't get away with saying that, due to the odd property of the integrated function, the result is $0$ (an obvious example is $1/x$). Do I need to work out the whole thing or is there a way to prove that odd functions with no singularities actually do integrate to $0$ over improper integrals.
I looked around but I couldn't find any mention about it, though in one of my books they reduce a similar integral to zero without explaining, hence my question.