I want to show asymptotics of the following integral involving Legendre Polynomial:
For $0<t<\theta<\frac\pi2$, $$\Big|\int_0^\theta \frac{1}{\sqrt{\theta}+\sqrt{\theta-t}} \frac{1-P_n(\cos t)}{t}dt \Big| \leq C \theta^{-\frac12} \cdot \log (n\theta),$$ where $C$ is the constant, $P_n(x)$ is a Legendre Polynomial, $n$ is a positive integer.
I am trying to use Bernstein's inequality for trigonometric polynomials:
Since $P_n(1)=1$ and $|P_n(\cos t)|\leq 1$, $$|P_n(\cos t)-1|=|P_n(\cos t)-P_n(\cos 0)|\leq nt||P_n ||_{\infty}\leq nt.$$
Then I can only get the below estimate but I have no further idea $$\Big|\int_0^\theta \frac{1}{\sqrt{\theta}+\sqrt{\theta-t}} \frac{1-P_n(\cos t)}{t}dt \Big| \leq \int_0^\theta \frac{1}{\sqrt{\theta}+\sqrt{\theta-t}} \frac{|1-P_n(\cos t)|}{t}dt \leq \int_0^\theta \frac{1}{\sqrt{\theta}+\sqrt{\theta-t}} \frac{nt}{t}dt=O(n).$$
Any suggestions are welcome! Thank you for your help!