Derive expressions for the angular momentum and kinetic energy of a rigid body in terms of its mass $M$, the position $\mathbf{X}(t)$ of its centre of mass, its inertia tensor $I$ (which should be defined) about its centre of mass, and its angular velocity $\boldsymbol{\omega}$.

A spherical planet of mass $M$ and radius $R$ has density proportional to $r^{-1} \sin (\pi r / R)$. Given that $\int_{0}^{\pi} x \sin x d x=\pi$ and $\int_{0}^{\pi} x^{3} \sin x d x=\pi\left(\pi^{2}-6\right)$, evaluate the inertia tensor of the planet in terms of $M$ and $R$.