Elliptic integral

1. The problem statement, all variables and given/known data

The problem is to calculate integral [tex]\int_{0}^{\pi/2}\frac{dx}{\sqrt{\sin{x}}}[/tex] by transforming it into elliptical form (complete elliptical integral of first kind).
First substitute a new variable theta with sin(x) = cos(theta). Then substitute a new variable phi with theta = 2 phi. Then you should have:

sed to generate this LaTeX image:

-2\int_{0}^{\pi/4}\frac{d\phi}{\sqrt{\cos{ 2\phi}}}

Now use the double angle formula for cosine given by cos(2a) = 1 - 2 Sin(a)^2 and you should be home free.
Thanks. I get it to the form [tex]2\int_{0}^{\pi/4}\frac{d\phi}{\sqrt{1-2(sin\phi)^{2}}}[/tex], which in my opinion equals [tex]2F(\sqrt{2},\pi/4)[/tex], but according to Mathematica, the answer is [tex]\sqrt{2}K(1/2)[/tex].

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