Just thought Id check here to see if any one had the formula to project where a turning bullet will wind up, after a given time, speed, rate of turn, and angle.

This is not the answer, (consider it a bump) but on a quick search I found this Parabola calculator showing kicking a ball that may be good at math, but a bit of a fail on ball kick.

Click "kick it" and see how math simulates life. "Except for how the air affects it."

Since dt is not constant, you can't solve the integral up front. Nonetheless you can approximate it, if you can assume your game to run at a stable fps, because then you can basically see this function as

cos(A+B*x)*C where the integral would be sin(A+B*x)*C/B

x is your time

C = the constant dt you expect (60 fps -> 1/60) multiplied by the bullet speed

A = starting angle

B = angle turned per second

the result would be the x-position in relation to the starting point of your bullet

Maybe it is possible to eliminate the dt by adjusting the turnrates dynamically along the process, so that the function will look like this:

Since the motion is just along a circle we can calculate the center and radius of that circle and build a nice exact formula to find where the object will be. You do have to account for the case where the rate of turning 0, because motion will be in a straight line and my formula will fail.