Evaluate integral (Chern article)

It's not so scary, after all :) You're integrating the forms $\Phi_k$ over the unit sphere bundle at a fixed point $x_0$ of $M$ (your notation is different from his, since for him $M$ is the unit sphere bundle of the manifold $R$). For $k\ge 1$, the form $\Phi_k$ will involve at least one curvature form $\Omega_i^j$. The curvature forms are horizontal for the fibration $SM\to M$, and you're integrating over a fiber. So those integrals all vanish.