The Bethe-Salpeter equation is commonly used in quantum field theory and particle physics. However, it's not typically associated with standard numerical methods of integration, like the ones you might use for solving definite integrals. If you're referring to a specific numerical method, it might be a miscommunication or you're combining different concepts.

For the sake of clarity, let’s focus on the problem at hand: evaluating the definite integral of $\sin(x)$ from $\pi$ to $\frac{3\pi}{2}$. The integral is given as:

$I = \int_{\pi}^{\frac{3\pi}{2}} \sin(x) \, dx$

Step-by-step solution:

1. The antiderivative of $\sin(x)$ is $-\cos(x)$.
2. Apply the limits of integration:

$I = \left[ -\cos(x) \right]_{\pi}^{\frac{3\pi}{2}}$

3. Now, substitute the limits:

$I = -\cos\left(\frac{3\pi}{2}\right) + \cos(\pi)$

4. Evaluate the trigonometric values:

$\cos\left(\frac{3\pi}{2}\right) = 0 \quad \text{and} \quad \cos(\pi) = -1$

5. Therefore:

$I = -(0) + (-1) = -1$

Thus, the value of the definite integral is:

$I = -1$

Would you like more details or any further clarifications? Here are some related questions you might find useful:

1. What is the Bethe-Salpeter equation, and how is it used in physics?
2. How do you apply numerical integration methods like Simpson's rule or trapezoidal rule?
3. What are common methods of solving definite integrals when no elementary antiderivative exists?
4. How do you evaluate integrals involving trigonometric functions with non-standard limits?
5. How does the periodicity of trigonometric functions affect definite integrals?

Tip: When integrating trigonometric functions over their periodic intervals, always check for symmetry, as it can simplify calculations.