doing my homework for a Machine Learning course, I have to calculate the following expression:
$\newcommand{\IDENTITY}{\mathbf{I}} \newcommand{\W}{\mathbf{w}} \newcommand{\WT}{\mathbf{w}^T} \newcommand{\PHI}{\mathbf{\phi}} \newcommand{\PHIM}{\mathbf{\Phi}} \newcommand{\PR}{\prod_{n=1}^N} \newcommand{\PRM}[1]{\prod_{i=1}^M#1} \newcommand{\PRJ}[1]{\prod_{j=1}^M#1} \newcommand{\FRACBETA}{\frac{1}{\beta}} \newcommand{\HALF}{\frac{1}{2}} \newcommand{\T}{\mathbf{t}} \newcommand{\XM}{\mathbf{X}} \newcommand{\IDA}{\alpha^{-1}\IDENTITY} \newcommand{\IDB}{\beta^{-1}\IDENTITY} \newcommand{\DETIDA}{\alpha^{-M}} \newcommand{\DETIDB}{\beta^{-N}} \newcommand{\LIKELIHOODCONST}{\frac{1}{\sqrt{|2\pi\IDB|}}} \newcommand{\LIKELIHOODCONSTNICE}{ \left(\frac{\beta}{2\pi}\right)^{\frac{N}{2}} } \newcommand{\PRIORCONST}{\frac{1}{\sqrt{(2\pi)^M\left|\IDA\right|}}} \newcommand{\PRIORCONSTB}{\frac{1}{\sqrt{(2\pi)^M\DETIDA}}} \newcommand{\PRIORCONSTNICE}{\left(\frac{\alpha}{2\pi}\right)^{\frac{M}{2}}} \newcommand{\SUMWSQ}{\sum_{i=1}^M w_i^2} \newcommand{\LOG}[1]{\mathrm{log}\left(#1\right)} \newcommand{\EXP}[1]{\mathrm{exp}\left\{#1\right\}} \newcommand{\NORMAL}[1]{\mathcal{N}\left(#1\right)} \newcommand{\DATA}{\mathcal{D}} \newcommand{\PROB}[1]{p\left(#1\right)} \newcommand{\INTINF}[1]{\int_{-\infty}^{\infty}\left[#1\right]} \newcommand{\INTINFDW}[1]{\INTINF{#1}\mathrm{d}\W} \newcommand{\INTINFDWDT}[1]{\INTINFDW{#1}\mathrm{d}\T} \newcommand{\WTPHIEXPANDED}{\PRJ{\EXP{- w_j[\phi_n]_j)}}} \INTINFDW{ \PR\WTPHIEXPANDED \PRM{\EXP{-\frac{\alpha}{2}w_i^2}}} $
Where $\PHI_n$ is a vector and $[\PHI_n]_j$ means accessing its $j$-th element. Now, I don't know if my notation is correct, but the meaning of $\mathrm{d}\W$ is integration over all elements of $\W = (w_1,w_2,\ldots,w_M)$.
At this point my knowledge of calculus is a bit rusty. I don't know how to proceed. How do I integrate over such an arbitrary number of variables?