The integral presented is:

$\int_0^{\infty} \frac{e^{-t} \sin t}{t} \, dt$

This is a well-known improper integral, commonly evaluated using the technique of contour integration or by recognizing it as a form of the Dirichlet integral.

Solution:

1. This integral can be evaluated using the known result:

$\int_0^\infty \frac{e^{-t} \sin t}{t} \, dt = \arctan(1) = \frac{\pi}{4}$

This result follows from complex analysis or Fourier transform methods, but it is also recognized as a special case of the Laplace transform when $s = 1$.

Would you like a detailed step-by-step explanation of how to derive this result? Or do you want to explore an alternative method of solution?

Related Questions:

1. How can we apply contour integration to solve this integral?
2. What is the relationship between this integral and the Laplace transform of $\sin t$?
3. How does the improper nature of the integral affect its convergence?
4. Can we use numerical methods to approximate this integral? If so, how?
5. What are some other common improper integrals that resemble this form?

Tip: When dealing with improper integrals, always check for convergence before proceeding with the calculation.