Suppose I have a mutivariate Gaussian distributed variable $u\sim\mathcal{N}(\mu,\Sigma)$, where $\Sigma$ is a dense matrix. I wish to calculate the expectation of $f(u_i)$.
Is $$E(f(u_i))=\int f(u_i)\mathcal{N}(u_i|\mu_i,\Sigma_{ii})\,du_i$$ If so provide a simple proof? Also does this apply to other multivariate distributions? eg. Multivariate T.
Otherwise do I have to consider that $$E(f(u_i))=\int f(u_i)P(u_i|u_{\backslash i})\,du_i \,P(u_{\backslash i})\,du_{\backslash i}$$ where, $u_{\backslash i}$ indicates variable $u$ without the i-th dimension.
Aside: in my particular case $f(u_i)=\exp(-u_i)$, however I'm more concerned of the general case.