Transform integral of 1/√sin(x) into complete elliptical form

[psid]

Homework Statement

The problem is to calculate integral $$\int_{0}^{\pi/2}\frac{dx}{\sqrt{\sin{x}}}$$ by transforming it into elliptical form (complete elliptical integral of first kind).

 

[confinement]

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.

 

[psid]

Thanks. I get it to the form $$2\int_{0}^{\pi/4}\frac{d\phi}{\sqrt{1-2(sin\phi)^{2}}}$$, which in my opinion equals $$2F(\sqrt{2},\pi/4)$$, but according to Mathematica, the answer is $$\sqrt{2}K(1/2)$$.