What is the technique that should be used to evaluate $\int_{0}^{2\pi} (\sin (k\sin \theta))^2 \,d\theta\ $?

Question

$$\int_{0}^{2\pi} (\sin (k\sin \theta))^2 \,d\theta\ $$ Where $k$ is a real constant.

I searched for it and found out that it 'does not have an integral I tried using a substitution for sinθ but then both limits change to zero

expressible with elementary functions'. What does this mean? If so how to solve it?

Answer 1

Hint: $$\int_{0}^{2\pi} (\sin (k\sin \theta))^2 \,d\theta=\int_{0}^{2\pi} \dfrac12\left(1-\cos (2k\sin \theta)\right) \,d\theta$$ where $$\int_0^{\pi} \cos(x\sin t)\ dt=\pi J_0(x)$$ and $J_0$ is Bessel function of order $0$.