I am trying to determine the asymptotic expansion of the Legendre function as described in a Bender and Orszag text, but have been unable to - some solutions are online, but they approach it in different forms and I'd like to solve it using the below integral definition of the Legendre polynomial.
Specifically, defining:
$$P_n(x)=\frac{1}{\pi}\int_0^\infty(x+cos(\theta)\sqrt{x^2-1})^n d\theta $$
Show that for large n:
$$P_n(x) \sim \frac{1}{\sqrt{2\pi n}}\frac{(x+(x^2-1)^{1/2})^{n+1/2}}{(x^2-1)^{1/4}}$$
Any help is appreciated!