u = -x^2 + 10 x
phiD6[x_] = WaveletPhi[DaubechiesWavelet[6], x]

c = Integrate[u phiD6[x], {x, 0, 10}]

dw6 = Head[Simplify[WaveletPhi[DaubechiesWavelet[6], x], 0 <= x <= 11]];

dw6["InterpolationOrder"]

pts = Transpose[{Flatten[dw["Grid"]], dw["ValuesOnGrid"]}];

pw[u_] = Piecewise[{InterpolatingPolynomial[#, u], #[[1, 1]] <= u < #[[2, 1]]} &
                    /@ Partition[pts, 2, 1]];

Integrate[(10 x - x^2) pw[x], {x, 0, 10}]
   11.911233499644986

Total[Integrate[(10 x - x^2) InterpolatingPolynomial[#, x], {x, #[[1, 1]], #[[2, 1]]}] & /@
      Partition[Select[pts, 0 <= #[[1]] <= 10 &], 2, 1],
      Method -> "CompensatedSummation"]
   11.911233499644995

quad[{{x0_, y0_}, {x1_, y1_}}] =
Simplify[Integrate[(10 x - x^2) InterpolatingPolynomial[{{x0, y0}, {x1, y1}}, x],
                   {x, x0, x1}]]

Total[quad /@ Partition[Select[pts, 0 <= #[[1]] <= 10 &], 2, 1],
      Method -> "CompensatedSummation"]
   11.911233499644984

u = -x^2 + 10 x;
phiD6[x_] := WaveletPhi[DaubechiesWavelet[6], x];

NIntegrate[u*phiD6[x], {x, 0, 10}, Method -> "GaussKronrodRule", AccuracyGoal -> 6,
MaxRecursion -> 20, WorkingPrecision -> 10]

(* 11.91115887 *)