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The integral presented is:

\[
\int_{-\pi}^{\pi} \frac{\sin(n\varphi) \, d\varphi}{1 - 2a \sin(\varphi) + a^2}, \quad -1 < a < 1, \quad n = 0, 1, 2, \ldots
\]

### Solution Strategy:
This integral can be approached using contour integration or by recognizing it as a standard integral related to Fourier series. The presence of \(\sin(n\varphi)\) and the trigonometric expression in the denominator suggests that we might be dealing with a form of generating function for Chebyshev polynomials or a related function in complex analysis.

### Steps to Solve the Integral:

1. **Convert the Denominator Using Complex Numbers**: 
   Write \(\sin(\varphi)\) in terms of complex exponentials using:
   \[
   \sin(\varphi) = \frac{e^{i\varphi} - e^{-i\varphi}}{2i}
   \]
   and then rewrite the integrand.

2. **Simplify the Expression**: 
   Substituting this into the denominator will allow for a simplification using complex analysis, potentially turning the integral into a contour integral around the unit circle in the complex plane.

3. **Use Residue Theorem (if applicable)**: 
   Apply the residue theorem if the integrand is in a form that allows it, calculating the residues of the poles within the unit circle.

4. **Recognize Special Cases**:
   For specific values of \(n\) (like \(n = 0\)), the integral may simplify significantly. For other \(n\), standard results from integrals of trigonometric functions over symmetric limits like \(-\pi\) to \(\pi\) can be useful.

Would you like a detailed walkthrough of the complete calculation using complex analysis, or an approach based on properties of Fourier series?

Solve the integral \( \int_{-\pi}^{\pi} \frac{\sin(n\varphi) \, d\varphi}{1 - 2a \sin(\varphi) + a^2}, \quad -1 < a < 1, \quad n = 0, 1, 2, \ldots \).

This page explains how to solve the integral \( \int_{-\pi}^{\pi} \frac{\sin(n\varphi)}{1 - 2a \sin(\varphi) + a^2} d\varphi \) for values of \(n = 0, 1, 2, \ldots\). Using complex analysis techniques like the Residue Theorem and the Fourier series, this problem provides insight into trigonometric integrals and generating functions.

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