We have the equation of motion in polar coordinates:
$$\frac{d^{2}\vec r}{dt^2} = (\frac{d^2 |\vec r|}{dt^2} - |\vec r|\cdot (\frac{d\theta}{dt})^2)\hat r + (|\vec r|\cdot \frac{d^2\theta}{dt^2}+2\frac{d\theta}{dt}\cdot\frac{d|\vec r|}{dt})\hat \theta.$$
For example, we have a pendulum that is consisted by a rigid rod and mass and we wish to find its velocity at a certain angle after we release from and initial angle $\theta_0$.
We could solve the problem using the Work-Energy theorem but say we wish to solve it by the equation of motion. We would have to integrate it but I find some difficulties doing so, like in in the radial direction where you have to integrate the term $(\frac{d\theta}{dt})^2$. So how do we integrate it?
