Can you help me with this? I tried to apply partial integration but got no results; I also looked at a few tactics about the integrals of high-grade expressions, I couldn't find a stylish move.
$$\int_{0}^{2\pi}{{(1-\cos{(x)})}^{10}\cos{(10x)}}dx$$
$\int_{0}^{2\pi}{{(1-\cos{(x)})}^{10}\cos{(10x)}}dx$
definite integralsintegration
Best Answer
For $n \in \mathbb{N}$, the following holds: $$ \int_0^{2\pi}(1-\cos(x))^n\cos(nx)\,dx = (-1)^n\frac{\pi}{2^{n-1}}. $$ So in your case, you get $\frac{\pi}{512}$ as the solution. See here for a proof of this identity.