This is one possibility: use a function to print each value and write each value one element at a time:
a1 = OpenWrite["calc2.dat", FormatType -> OutputForm];
data = Table[OutputForm@Integrate[i*x^2, {x, 0, 1}], {i, 1, 6}];
Map[(Print[#]; Write[a1, #]) &, data];
Close[a1]
There are many ways to do this type of thing. But for large lists it might be better to write everything in one fell swoop, rather than item by item.
