Calculate $ \int_{0}^{\pi} \frac{dx}{\sqrt{3-\cos(x)}} $

Question

$$ \int_{0}^{\pi} \frac{dx}{\sqrt{3-\cos(x)}} $$

I need to calculate this using Beta \ Gamma functions. I have tried the substitution $2 +\cos(x) = t$

Answer 1

Let $x= 2t$ then we have that, $$\int_{0}^{\pi/2} \frac{2dt}{\sqrt{3-\cos(2t)}} = \int_{0}^{\pi/2} \frac{\sqrt{2}\cdot dt}{\sqrt{1+\sin^2(t)}}.$$ Then let $\sin(t)=u$ which gives, $$\int_{0}^{\pi/2} \frac{\sqrt{2}\cdot dt}{\sqrt{1+\sin^2(t)}}= \int_{0}^{1} {\sqrt{2}(1-u^4)^{-1/2}} du.$$ Let $u^4 = x$ then $$\int_{0}^{1} {\sqrt{2}(1-u^4)^{-1/2}} du=\frac{1}{2\sqrt{2}}\int_{0}^{1} x^{-3/4}(1-x)^{-1/2}dx.$$

Can you conclude from here?