For $k\in\mathbb{N}_{0}$ and $x\in\mathbb{R}$, define $$I_{k}(x):=\int_{0}^{\pi/2}\cos(xg(\theta))\sin^{2k}\theta\,\mathrm{d}\theta$$ where $$g(\theta)=\int_{\sin\theta}^{1}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-\alpha^{2}t^{2})}}$$ and $0<\alpha<1$ is a fixed parameter.

I arrived at the expression $I_{k}(x)$ while working on certain asymptotic analysis of Jacobi elliptic functions. I doubt that the integral $I_{k}(x)$ could be evaluated somehow (even in terms of some known special functions). Nevertheless, for my purpose, it would be sufficient to know the asymptotic behavior of $I_{k}(x)$, as $k\to\infty$. It seems, however, that for my calculations, I need to know not only the leading term of the asymptotic expansion, but the second term as well.

Does anybody know how to derive the asymptotic expansion for $I_{k}(x)$, as $k\to\infty$? Many thanks.