How Do I Integrate Nested Trigonometric Functions?

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SUMMARY

The integration of the function sin[y*sin(x)]*sin(x) with respect to x from -π to π presents significant challenges. The discussion confirms that the function is even, allowing the limits to be adjusted to 0 to π and then doubled. However, participants express skepticism regarding the existence of an elementary anti-derivative for this nested trigonometric function, suggesting that standard methods may not yield a solution. The mention of Bessel functions indicates a deeper mathematical context that complicates the integration process.

PREREQUISITES
  • Understanding of trigonometric functions and their properties
  • Familiarity with integration techniques, including integration by parts and substitution
  • Knowledge of even and odd functions in calculus
  • Basic concepts of Bessel functions and their applications
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  • Research advanced integration techniques for nested trigonometric functions
  • Study the properties and applications of Bessel functions
  • Explore numerical integration methods for functions lacking elementary anti-derivatives
  • Learn about special functions and their role in mathematical proofs
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Mathematicians, physics students, and anyone involved in advanced calculus or mathematical analysis, particularly those working with nested trigonometric functions and Bessel functions.

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How do I go about integrating

sin[y*sin(x)]*sin(x) wrt x from -pi to pi,

I've got that its an even function so I can change the limits to 0 to pi and double it, but I can't find the analytic answer. By parts? substitution? although for substitution I assume there needs to be a cos function somewhere. Any ideas?

Thanks
 
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So y is a constant here?

Do you have any reason to think that there IS an elementary anti-derivative? (Most functions do not.)
 
Actually, I have no reason to believe it is doable. It came up in a proof I was working on involving some bessel function. I was just wondering if their was a standard method for nested trig functions.

Thanks for your time
 

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