How can I calculate $\int_{-\pi}^{\pi} \cos^6(x)dx$ by using Parseval's theorem

Question

How can I calculate by using Parseval's theorem:

$$\int_{-\pi}^{\pi} \cos^6(x)dx$$

Answer 1

Take $x(t) = \cos^3(x)$. And express it in terms of $\cos(nx)$ by application of Double-angle/Trigonometric Addition formulas.

You should find that the coefficients of $\cos(nx)$ at $n = 1, 3$ are $a_1 = 0.75$ and $a_3 = 0.25$, the rest vanish.

Apply Parseval: $\frac{1}{\pi}\int_{-\pi}^{\pi} \cos^6(x) = \sum |a_n|^2$.