Can we find the integral $\int_{0}^{2\pi} \cos(\cos x)\,dx$?

Question

Can we find the integral $$\int_0^{2\pi} \cos(\cos x)\,dx\;?$$

Answer 1

For first, consider that: $$ \int_{0}^{2\pi}\cos^{2n}x\,dx = \frac{2\pi}{4^n}\binom{2n}{n}\tag{1} $$ hence by exploiting: $$ \cos z = \sum_{n\geq 0}\frac{(-1)^n z^{2n}}{(2n)!}, \tag{2}$$ replacing $z$ with $\cos x$ and integrating termwise we get: $$ \int_{0}^{2\pi}\cos(\cos x)\,dx = 2\pi\sum_{n\geq 0}\frac{(-1)^n}{4^n(n!)^2}=\color{red}{2\pi\, J_0(1)}. \tag{3}$$ Look at this Wikipedia page for the definition of the Bessel function $J_0$.

Answer 2

$$\int_0^\tfrac\pi2\sin(a~\sin x)~dx~=~\int_0^\tfrac\pi2\sin(a~\cos x)~dx~=~\frac\pi2~H_0(a)$$

$$\int_0^\tfrac\pi2\cos(a~\cos x)~dx~=~\int_0^\tfrac\pi2\cos(a~\sin x)~dx~=~\frac\pi2~J_0(a)$$

See Bessel and Struve function for more information. Also,

$$\int_0^\tfrac\pi2\cos(a~\tan x)~dx~=~\int_0^\tfrac\pi2\cos(a~\cot x)~dx~=~\frac\pi2~e^{-|a|}$$