How to solve a challenging integration problem?

[Pi314]

I am stuck on an intergration and can't get a simple answer which i should get

p = (2*m*l^2)^(1/2)*(E+m*g*l*cos theta)^(1/2)

using small angle (need for question) cos theta = 1 - (theta^2 )/2

Then I intergrate from 0 to 2*Pi and get some very large eqn using a maths program, I can't think how I would do it by hand?

Can anyone help with the problem

 

[HallsofIvy]

So basically, the problem is to find the integral (a+ bcos(theta))^1/2? That looks like an "elliptic integral" to me- there is no way to do it in terms of elementary functions.

You mention the "small angle" approximation but, of course, 2pi is NOT a "small angle".