An Elliptic Integral

Date: 01/05/2003 at 08:52:03
From: Rajiv (i am in college)
Subject: Integration

Integrate Sqrt[Sin[x]]

I tried to solve this problem using the usual approaches, but they do 
not seem to work. It seems that the problem can be solved only by 
applying some trick or technique. One of my friends suggested that it 
can be solved by using Laplace transform, but I could not solve it 
that way.

Date: 01/07/2003 at 14:09:19
From: Doctor Wilkinson
Subject: Re: Integration

Interesting problem, Rajiv!  Doing the substitution t = tan(x/2) 
reduces this integral to the integral of 

  (2t/(1+t^2))^(1/2) (1+t^2)^(-1) 

and this is an elliptic integral. It cannot be expressed in terms of 
elementary functions, but can be expressed in terms of the inverses of 
the Jacobian elliptic functions (with a bit of work). The same applies 
to the integrals of the square root of cosx or tanx.

- Doctor Wilkinson, The Math Forum
  http://mathforum.org/dr.math/