The image you uploaded contains the following integral:

$I = \int_{-\infty}^{\infty} \frac{x \sin(\pi x)}{x^2 + 2x + 17} \, dx$

This is an improper integral of a rational function involving trigonometric terms. To solve it, we can employ techniques such as contour integration or the use of known integral results, as this is a standard form often encountered in Fourier analysis or complex analysis.

Would you like a step-by-step explanation of how to approach this integral, or do you have a specific method you'd like to discuss? Let me know if you'd like further details.

Here are some related questions that could expand on this topic:

1. What is the method of contour integration, and how does it apply to integrals like this?
2. How can we evaluate integrals involving trigonometric functions and polynomials in the denominator?
3. What are some standard integral formulas that might help with integrals of this form?
4. How do you determine if an improper integral converges or diverges?
5. What is the role of the Fourier transform in evaluating integrals like these?

Tip: To solve integrals involving trigonometric terms, it can be useful to use symmetry or convert to a complex function to take advantage of residue calculus.