How to integrate a normalized polyfit symbolically over the original X-values?

Hi Guys,

I have a residual stress distribution over a plastically bent beam as measurement data. This stress distribution must be integrated between zero and a certain crack length a, with x as integration variable.

I used a 12th order (large data set) polyfit to fit the data, but the equation is badly conditioned. To solve this, I normalized the x-data by

x_norm = (x - mean(x)) / std(x);

to get

sigma = p1 * x^12 + p2 * x^11 + ... + p13

This is all okay and the X-values are now in the range of [-1.5,1.5] but I need to integrate this polynomial symbolically (syms x) over the original crack length with x between 0 and a.

Any ideas?