Let $p\in \mathbb{N}.$ I would like to know how to compute $$\int_{0}^{2\pi} \cos^j(t) \cos(kt) dt, \quad \quad j,k\in \{1,2,\ldots,p\}.$$ Could somebody help me with this? It might be related to complex analysis (computing residue) or Fourier analysis. Thank you so much.
Mana
Use \begin{align} \cos^j(t)&=\frac{(e^{it}+e^{-it})^j}{2^j} \\&=\frac1{2^{j-1}}(\cos(jt)+\tbinom{j}{1}\cos((j-2)t)+\tbinom{j}{2}\cos((j-4)t)+…) \end{align} where the last term is $\frac12·\binom{j}{j/2}$ for even $j$ and $\binom{j}{(j-1)/2}\cos t$ for odd $j$.
Then apply what you know about the formula for the Fourier coefficients.