Derivation of Rotational Motion Equations using Calculus

Calculate the positions $r(t)$ of any three distinct, non-coplanar points A, B, C in the rigid body. As the object translates and/or rotates, all other points in the rigid body maintain the same positions relative to A, B, C.

If $alpha$ is a constant then the orientation of the object in 2D is $theta=frac12 alpha t^2$.

You don't need those general linear-motion definitions/equations. Rather, you need the equivalent rotational-motion definitions/equations:

$$mathbf{omega}(t) = mathbf{omega}_0+int_{t_0}^t mathbf{alpha}(t')dt' mathbf{theta}(t) = mathbf{theta}_0+int_{0}^t mathbf{omega}(t')dt'$$

These are fundamental definitions that just mathematically reflect the fact that angular velocity is the change in angular position, and angular acceleration the change in angular velocity, in the same way as in the linear case.

The four usual motion equations are derived by assuming $alpha$ constant, and in exactly the same way as the linear motion equations are derived. See a derivation here.