COnverting integral into First Order of Bessel Fuuction of first kind

Question

How to prove

$$ \frac {\omega^2 \int_0 ^{2\pi/\Omega} \sin \left(\Omega s\right) \sin \left(A \cos \left(\Omega s\right) \right)ds}{\int_0 ^{2\pi/\Omega} A \sin \left(\Omega s \cos \left(\Omega s \right)\right)ds} = \frac {2 \omega^2 J_1 \left(A\right)}{A}, $$

with $\omega, \Omega, A$ being constants, where $J_1(A)$ stands for the first order Bessel function of the first kind.

Thank you.

Answer 1

first, i think that i can change the left hand side form into $$\frac{{\omega^2 \int_0 ^{2\pi/\Omega} 2 \sin \left(\Omega s\right) \sin \left(A \cos \left(\Omega s\right) \right)ds}}{{\int_0 ^{2\pi/\Omega} 2A \sin \left(\Omega s \cos \left(\Omega s \right)\right)ds}}$$ but the denumerator result is zero, so the equation will be undefined. and now i still seek an idea to decompose the integral above into first order of bessel equation of the first kind.