Calc $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty -\frac{t}{1+t^2}(\delta (\omega-t-\pi)-\delta(\omega-t+\pi))dt$

Question

The answer to this integral:$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty -\frac{t}{1+t^2}(\delta (\omega-t-\pi)-\delta(\omega-t+\pi))dt$$

is $$\frac{1}{\sqrt{2\pi}}\left(-\frac{\omega-\pi}{1+(\omega-\pi)^2}+\frac{\omega+\pi}{1+(\omega+\pi)^2}\right)$$ It came from a Fourier transform convolution integral.

I have no idea how to compute it. Help?

Answer 1

Hint: remember that $$ \delta(x)=\left\{\begin{array}{rl} 1&\text{ if }x=0,\\ 0&\text{ else.} \end{array}\right. $$