1. ## residues, improper integral

Use residues to evaluate the improper integral.
$
\int^{\infty}_{-\infty} \frac{\cos x dx}{(x+a)^2 + b^2}$
( $b>0$).

I do not see how to evaluate this integral now. In this section, we use $M_R$ and $C_R$. I do not see how to use this here. I need hints on how to do this. Thanks in advance.

2. Originally Posted by canberra1454
Use residues to evaluate the improper integral.
$
\int^{\infty}_{-\infty} \frac{\cos x dx}{(x+a)^2 + b^2}$
( $b>0$).

I do not see how to evaluate this integral now. In this section, we use $M_R$ and $C_R$. I do not see how to use this here. I need hints on how to do this. Thanks in advance.
You do understand, don't you, that " $M_R$" and " $C_R$" are not universal symbols? I have no idea what you mean by that.

But what I would do is set up a contour that runs along the real axis from -R to R, then along the semi-circle in the upper complex half-plane back to R.

It should be easy to show that, as R goes to infinity, the the integral around the semi-circle goes to 0 so the integral on the real axis is equal to the residues in the upper complex half-plane. And in that half-plane your integrand is analytic everywhere except where $(x-a)^2+ b= 0$; in other words at $x= a+ i\sqrt{b}$. Find the residue at that point.