Of course you can do integrals in POV-ray. You can do anything with POV-ray.
*Fanatical glint in the eye*
You can obtain a numerical approximation of the integral by applying its
definition, i.e. take a very small dx, evaluate y and add dx*y to previous
sum in the loop.
You can also use the generalised hypergeometric solution, see
http://mathworld.wolfram.com/CornuSpiral.html The generalised
hypergeometric looks like a nasty customer to code, but it has the
advantage of being an exact solution.

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