How to Solve Complex Integrals in Lagrangian Mechanics?

[Kuryakin]

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.

$$<br /> <br /> \int_{0}^{x}dx(\frac{xsin(x)}{1+kcos^2(x)}})^\frac{1}{2}<br />$$ Any hints? I'm not sure where to get started.

 

[dextercioby]

It's a typical elliptic integral. It can be solved only approximately.

 

[Kuryakin]

Thanks. I'll have a go at the taylor expansions later.

 

[Kuryakin]

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.

 

[Kuryakin]

$$\int_{0}^{x}\frac{xsin(x)dx}{1-kcos^2(x)}}<br />$$

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