Simple cosine expression NOT so simple

[Mark Wood]

I am rediscovering my long lost "A" level maths and have been having lots of fun working on puzzles. Unfortunately, I seem to have hit something of a wall in the following simple expression that is repelling every attack I launch:

f(x)=Sqrt(1+kcos(x))

I have tried substituting x=arccos(w) and w=arccos(w/k), expressing things as logs, mungeing the equations I get, and a number of slightly more desperate attacks, but whatever I do to get rid of one term or provide something that can be canceled out, a new problem pops up.

I feel that there is a simple answer to this, but the more I look at it the less I can see the way forward.

Any suggestions anyone?

 

[Dick]

What exactly do you want to do with f(x)?

 

[Mark Wood]

Ah! So absorbed by it, I've missed the most important bit!

I'm trying to integrate it to get the area under the curve between x=0 and x=pi

 

[Sourabh N]

Sorry, i couldn't get ur expression. is it (1+kcos(x))^(1/2)

 

[Mark Wood]

Yes it is - the root of (1+kcos(x))

 

[neutrino]

I don't think it has a solution made up entirely of elementary functions. Elliptic functions may be involved.

 

[Sourabh N]

I too struggled for it for 6-8 months. Its a kind of elliptic integral(which i didn't knew). u might be knowing. u can check it on an integrator at www.wolfram.com

 

[Mark Wood]

Gosh - what a great tip. I didn't know the integrator existed. Thank you.

Having got the answer from Wolfram I'm kind of pleased that I hadn't missed an obvious solution.

Thank you all.