Precision and time of calculating the integral in Mathematica

1)How to calculate this integral, because I have some very small values (10^-20) in solution and values on this position in matrix should be equal to zero. If I can not do that with working precision, how to set automatically this values to be zero in matrix for further?

2)Also there is a problem, because I don't know what is a true. Maybe mathematica is wrong because of precision, if it is not (I suppose), I want to set this values to be zero.

3)This command is not numerical integration, but can I use parallelize or other ways to obtaining the solution faster?

Here is a file:

www.sendspace.com/file/q7xukg

Thank you very much in advance

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1. Try changing the bounds of your integration so that it uses exact numbers (1/5, 1/2, etc) instead of .2, .5, etc. If you do this Mathematica will integrate your function exactly (it's just a polynomial, after all.) Alternatively, you can specify that the integration bounds have, say, 30 digits of precision by using .230 (that's a backtick) instead of .2`. If you don't, Mathematica will assume the .2 has machine precision, and will only calculate the integral with machine precision... asking for 30 digits of precision in the answer after the answer has already been computed will then not help.