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 Is it possible to do the eulicidean distance of the acceleration, subtract the 9.81, then do the integration to get the distance travelled? Is this valid? Would it work?
If what you have is the magnitude of the net acceleration then integrating it will not give you a distance travelled. Think about a centrifuge -- high acceleration but net distance travelled is negligible.

What you need to do is to subtract the 9.81 from the "z" coordinate of the acceleration and then integrate as before. If your z axis is not perfectly vertical then you would want to convert that 9.81 to an (x,y,z) acceleration value and subtract that from x, y and z.