The Bethe-Salpeter equation typically refers to a framework in quantum field theory, primarily used to describe bound states like mesons in quantum chromodynamics (QCD). However, it doesn't traditionally apply to methods for evaluating definite integrals. This method is not typically a recognized numerical technique for integrating simple functions like $\sin(x)$.

Instead, numerical methods for integration commonly include Simpson's rule, Trapezoidal rule, Gaussian quadrature, or others. If you're interested in exploring numerical integration methods specific to physics or quantum field theory, those typically belong in a different context.

If you'd like, I can guide you through a proper numerical integration method for evaluating $\int_{\pi}^{3\pi/2} \sin(x) \, dx$.

Would you like to proceed with a specific method, such as Simpson's rule or Trapezoidal rule? Feel free to upload any related materials if needed.