What is the Stepwise Solution for Solving a Contour Integration Problem?

[kranav]

Hello! I wanted to solve this integral but really didn't understand the method show in the book.
Can anyone help me out please.

sorry I don't know how to show the integral sign, here it is

integral of - to + infinity (Cos(x)/x^2 + 2x +5 )dx

here Cos(z) = Re[exp(iz)]

I tried to reduce the denominator to a (a+b)^2 thing and then use a method that I didn't understand ( so I copied it from the book).
I need to know the stepwise solution if possible.

Thank You!

 

[HallsofIvy]

I presume you mean you wrote $x^2+ 2x+ 5= x^2+ 2x+ 1+ 4= (x+1)^2+ 4$. The function $e^z/((z+1)^2+ 4$ is analytic everywhere except at z= -1+2i and -1-2i where there are simple poles.

If you take your contour to be along the x-axis from (-R, 0) to (R, 0), then around the half circle in the upper half plane from (R, 0) to (-R, 0), the integral is the residue at -1+ 2i divided by $2\pi i$. The integral you want is the integral on the x axis, as R goes to infinity. If you can find the integral around $z= Re^{i\theta}$ as $\theta$ goes from 0 to $\pi$, you can subtract that from the inegral around the contour, for any R, to get the integral along the x-axis..

 

[kranav]

thank you very much.