How to Solve Complex Integrals in Lagrangian Mechanics?

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Homework Help Overview

The discussion revolves around solving a complex integral related to Lagrangian mechanics, specifically involving elliptic integrals. The original poster expresses difficulty with an integral that has a constant parameter affecting its solvability.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the nature of the integral, identifying it as a typical elliptic integral that requires approximate methods for solutions. There are attempts to use Taylor expansions and numerical methods like Simpson's rule to evaluate the integral.

Discussion Status

The conversation is ongoing, with participants exploring different approaches to tackle the integral. Some guidance has been offered regarding the use of Taylor series and numerical methods, but there is no explicit consensus on a definitive solution approach yet.

Contextual Notes

Participants are considering the implications of the constant k being close to 1, which may affect the methods used for approximation. The original poster's experimental setup imposes constraints on the solution's accuracy.

Kuryakin
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Homework Statement



Trying to solve a Lagrangian, got down this integral. Unfortunately the zeroth-solution isn't good enough since the constant k is close to 1 for our experimental set-up.

[tex] <br /> <br /> \int_{0}^{x}dx(\frac{xsin(x)}{1+kcos^2(x)}})^\frac{1}{2}<br /> [/tex] Any hints? I'm not sure where to get started.
 
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It's a typical elliptic integral. It can be solved only approximately.
 
Thanks. I'll have a go at the taylor expansions later.
 
Had an attempt at the Taylor series and it looked like it was just going to make it more complex. I've plotted it in excel and get a nice semi-ellipse then I'll just use simpsons rule to solve for the area.
 
[tex]\int_{0}^{x}\frac{xsin(x)dx}{1-kcos^2(x)}}<br /> [/tex]

Looks a little more friendly. Is this still only solvable by approximate methods?
 

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