What is the closed-form of $\int_{0}^{\pi}\text{sin}(n\tau-x\text{sin}\tau)d\tau$

Question

I know the Bessel function can be formatted as follows. $$J_n\left(x\right)=\frac{1}{\pi}\int_{0}^{\pi}\text{cos}\left(n\tau-x\text{sin}\tau\right)d\tau$$ So can we get a Bessel-similar expression for the integration in the title? Many thanks!

Answer 1

According to https://en.wikipedia.org/wiki/Anger_function, https://dlmf.nist.gov/11.10,

$\int_0^\pi\sin(n\tau-x\sin\tau)~d\tau=\pi\mathbf{E}_n(x)$