Decided to try an experiment
In[1]:= f[x_] = Sin[x]*Sin[3 x];
In[10]:= NIntegrate[f[x], {x, 0, 10}]
Out[10]= 0.135097
In[12]:= pts = Sort[RandomReal[{0, 10}, 100]];
In[14]:= data = Table[{pts[[i]], f[pts[[i]]]}, {i, 100}];
In[21]:= i[n_] := Interpolation[data, InterpolationOrder -> n]
In[24]:= Off[InterpolatingFunction::dmvali]
In[25]:= Table[Integrate[i[n][x], {x, 0, 10}], {n, 0, 3}]
Out[25]= {0.154373, 0.0429017, 0.163184, 0.181286}