write integral SQRT(cosx) in terms of elliptic integrals

[dts5044]

hey. I have a problem in calculus I can't seem to find a starting point on. We're working on elliptic integrals and the problem is:

write [integral from 0 to pi/2] SQRT(cosx)dx in terms of the elliptic integrals E and F

where E(k,q) = [integral from 0 to q] SQRT( 1 - (k^2)*(sinx)^2)dx

and F(k,q) = [integral from 0 to q] dx / (SQRT(1 - (k^2)*(sinx)^2)

so far I've changed SQRT(cosx) to (1-sinx^2)^(1/4) ...but I can't see where that would lead. A starting point would be good!

 

[royhaas]

Another identity is \(\displaystyle \cos(x) = 1-2\sin^2(x/2)\).

 

[dts5044]

hmmm...i'm still not sure where to go from there. I forgot to mention 0 < k < 1...maybe that substitution is still helpful, but if you were thinking to simply substitute and then the problem is over, i dont think that's it. also 0 <= q <= pi/2

since the problem wants the solution in terms of both E and F, is it going to eventually involve some sort of integration by parts?