$\frac{1}{\pi} \int_{0}^{2\pi} \theta^m \exp\left(-\sum_{k=1}^{K} x_k e^{-ik\pi\sin(\theta)}\right) d\theta$

Question

Introduction

Seeking insights on generalizing the integral solution from

$$\frac{1}{\pi} \int_{0}^{2\pi} \exp\left(-\sum_{k=1}^{K} x_k e^{-ik\pi\sin(\theta)}\right) d\theta$$Solution is here

to

$$\frac{1}{\pi} \int_{0}^{2\pi} \theta^m \exp\left(-\sum_{k=1}^{K} x_k e^{-ik\pi\sin(\theta)}\right) d\theta$$

Questions

1. Simplification Methods: Are there techniques to simplify the analysis?
2. Solution Insights: Can the first integral's solution guide solving the second? especially for $m = 1$ and $m = 2$.
3. Literature References: Any similar forms or strategies discussed in literature?

Context

This inquiry stems from research in wave propagation and Fourier series, where such integrals are common.

Request

Appreciate any guidance or references. Thank you!