I have to estimate the following integral $$\left|\int_{\frac{\pi}{k}}^{\frac{\pi}{2}} (\sin \theta)^{-1+\frac{i(n+1)y}{2}} d\theta \right|,\quad \forall k,n\geq 2 $$

According to Sogge (Oscillatory Integrals and Spherical Harmonics. Duke Mathematical Journal 53 (1986), no. 1, 43-65) the integral is $\leq C \max(1,|y|^2)$ where $C$ is independent of $y,k$, and this can be proved by a routine integration by parts argument but I cannot figure it out.

Thanks in advance!


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