I am investigating the asymptotic behavior of the integral $\int_0^{\theta} \sin^n x \, dx$ for any positive integer $n$ and $\theta \in [0, \pi]$. Current research, such as the discussion on Asymptotic properties of the integral of the power of the sine function, primarily focuses on asymptotic properties. In that thread, @Mariusz Iwaniuk proposed the following approximation:$$ I_n \sim \frac{\sqrt{\frac{\pi}{2}} \sin^{n+1}(\theta) e^{\frac{1}{2} n \cos^2(\theta)} \operatorname{erfc}\left(\frac{\sqrt{n} \cos (\theta)}{\sqrt{2}}\right)}{\sqrt{n}} $$I am very interested in understanding the proof of this expression. Any theoretical guidance or references that could assist would be greatly appreciated.
[Edited] Many thanks to @Gary for providing an asymptotic bound! To normalize, I've scaled the integral under investigation by the integral over $[0, \pi]$, which is:$$ \frac{\int_0^{\theta} \sin^n(x) \, dx}{\int_0^{\pi} \sin^n(x) \, dx} $$I compared the bounds provided by Mariusz Iwaniuk and Gary. Limited simulations indeed confirm that Gary’s appears to be an upper bound and Mariusz Iwaniuk’s a lower bound. Particularly, Gary's upper bound aligns closely across $\theta \in [0, \pi]$, whereas Mariusz Iwaniuk’s accuracy decreases from $\pi/2$ to $\pi$, likely due to a higher power of $\sin \theta$ in his formula. My current challenge is to obtain proofs for these bounds and to improve the lower bound approximation between $\pi/2$ to $\pi$. The simulation results are as follows:result