NN = 300; n = 30; M = 5;
k = a*(M + 1)^2/NN;
ei[l_] := 1 + 2*k*Cos[Pi*l/(M + 1)] - 2*k;
prod = 1;
For[l = 1, l <= M, l++,
  prod = prod*ei[l];
  ];
dA = Simplify[prod];
R = ConstantArray[0, {M, M}];
For[jj = 1, jj <= M, jj++,
For[l = 1, l <= M, l++,
    R[[jj, l]] = Sin[l*Pi*jj/(M + 1)];
    ];
  ];
Q = Simplify[Sqrt[2/(M + 1)]*R];
Dinvn = ConstantArray[0, {M, M}];
For[l = 1, l <= M, l++,
  Dinvn[[l, l]] = 1/ei[l]^n;
  ];
Ainvn = Q.Dinvn.Transpose[Q];
dAi = 1/dA^n;
f[x_] := PDF[
   ProductDistribution[NormalDistribution[0, 1],
   NormalDistribution[0, 1], NormalDistribution[0, 1],
   NormalDistribution[0, 1], NormalDistribution[0, 1]], x];
w = {x[1], x[2], v, x[4], x[5]};
integrand = f[Ainvn.w]*dAi;
NIntegrate[integrand /. v -> 1, {x[1], -[Infinity], [Infinity]},{x[2],-[Infinity],[Infinity]}, {x[4], -[Infinity], [Infinity]}, {x[5],-[Infinity], [Infinity]}, {a, 1, 2}]

NIntegrate::izero: Integral and error estimates are 0 on all integration subregions. Try increasing the value of the MinRecursion option. If value of integral may be 0, specify a finite value for the AccuracyGoal option.

0.

dAi /. a -> 1.4

2.06591*10^30

Ainvn.w /. {x[1] -> 1, x[2] -> 1, x[4] -> 1, x[5] -> 1, a -> 1.5, v -> -0.5}

{-6.71576*10^13, 1.1632*10^14, -1.34315*10^14, 1.1632*10^14, -6.71576*10^13}