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式を統合する場合、n個の変数とn次元のnに依存し、nも変数になります。
このようなもの?
multyIntegrate[fun_, n_Integer] := With[{x = Sequence @@ Table[Unique[mi], {n}]}, Integrate[fun[x], x]] multyIntegrate[g, 3]
f[i___] := Total[{i}.{i}] multyIntegrate[f, 3]