How do I go about finding the principal value of this integral? $$\int_{-\infty}^{\infty}\frac{\sin(x)}{a-x}dx$$ Another post suggested that shifting the integrand by making the change $x \rightarrow x+a$ is one way to simplify to two easier integrals. Can someone help me solve this without changing the integrand like that with a more direct approach (a summary of it will suffice)?
Thank you