Calc $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty -\frac{t}{1+t^2}(\delta (\omega-t-\pi)-\delta(\omega-t+\pi))dt$ 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?
Hint: remember that 
$$ \delta(x)=\left\{\begin{array}{rl}
1&amp;\text{ if }x=0,\\
0&amp;\text{ else.}
\end{array}\right. $$
Hint: remember that 
$$ \delta(x)=\left\{\begin{array}{rl}
1&amp;\text{ if }x=0,\\
0&amp;\text{ else.}
\end{array}\right. $$