I need to find the value of the following integral using complex analysis:
$$\int_{-\infty}^{\infty}\frac{\sin(k_1\ x)+\sin(k_2\ x)}{x^2-a^2}\ dx$$ where $k_1, k_2, a$ are real coefficients.
The poles are obviously $\pm a$, so should I apply the residue theorem next?
I get the answer $0$ (which should be correct because the function is even) in a couple of steps, but I never used the condition that the coefficients are real and I'm not sure if my chosen contour of integration is correct.
Any help?
For starters, the integral does not converge near $a$ and $-a$. You can take the Cauchy Principal Value, but that is not the same as converging. The Cauchy Principal Value is $0$, by the way, and that is because the integrand is odd.