Elliptic Integral Homework: Calculate \int_{0}^{\pi/2}\frac{dx}{\sqrt{\sin{x}}}

[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:


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

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).