¿Qué es un buen destino a $\int^{2\pi}_{0}\cos^k(x)dx$?

Question

Integral $$\int^{2\pi}_{0}\cos^k(x)dx$$comes up in one of Rudin's problems. He asked me to do a detailed approximation of it. I am wondering how to do it. This feels really elementary, but I need a lower bound that works for any $$%k. Integración por partes no realmente ayudaría a pesar de que la función es integrable explícitamente.