How to solve a challenging integration problem?

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SUMMARY

The discussion focuses on solving a challenging integration problem involving the expression p = (2*m*l^2)^(1/2)*(E+m*g*l*cos(theta))^(1/2). The user applies the small angle approximation, cos(theta) = 1 - (theta^2)/2, to simplify the integral from 0 to 2*Pi. However, they encounter difficulties in calculating the integral of (a + bcos(theta))^(1/2), which is identified as an elliptic integral, indicating that it cannot be expressed in terms of elementary functions. The user seeks assistance in performing this integration by hand.

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  • Understanding of integral calculus, particularly integration techniques.
  • Familiarity with elliptic integrals and their properties.
  • Knowledge of small angle approximations in trigonometry.
  • Experience with mathematical software for symbolic computation.
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  • Research methods for solving elliptic integrals, specifically the integral of (a + bcos(theta))^(1/2).
  • Learn about the application of small angle approximations in physics and engineering problems.
  • Explore advanced integration techniques, including numerical methods for complex integrals.
  • Investigate mathematical software options, such as Mathematica or MATLAB, for symbolic integration.
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This discussion is beneficial for mathematicians, physics students, and engineers dealing with complex integration problems, particularly those involving elliptic integrals and approximations in trigonometry.

Pi314
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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
 
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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".
 

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