The problem in the image asks to determine whether the integral

$\int_{-\infty}^{0} e^x \sin(x) \, dx$

converges or diverges.

To determine this, we can apply integration techniques such as integration by parts or analyze the behavior of the integrand as $x \to -\infty$. Given that the integrand involves an exponential function and a trigonometric function, integration by parts can help simplify the problem.

The solution method provided in the image indicates the use of integration by parts, which is typically a standard approach when dealing with products of exponential and trigonometric functions. The integral might also require applying integration by parts twice.

Would you like a detailed step-by-step explanation of how to solve this integral, or do you need further clarification on any part of the process?

Here are 5 related questions you might find helpful:

1. What is the formula for integration by parts?
2. How does the exponential function $e^x$ behave as $x \to -\infty$?
3. What are the conditions under which an improper integral converges?
4. Can you solve a similar integral where the limits are from 0 to infinity?
5. How would you approach an integral involving a product of trigonometric functions and polynomials?

Tip: When integrating functions like $e^x \sin(x)$, it's often helpful to remember that integration by parts is a powerful tool, especially when the integrand is a product of a simple exponential function and a trigonometric function.