Let $\lambda\in\mathbb{R}$ and consider the integral $$\int_{[-\pi,\pi]} \frac{|\sin(x)|^{-\lambda i}}{x}dx.$$ I'd like to approximate it as $\lambda\to\infty$. Without the singularity in $x=0$, that would be done using the theory of stationary phase. However, I'm not sure how to address the contribution of the pole at $x=0$ to this integral.
How can I get the asymptotic for this integral?