# Residue theorem for real integrals

1. Sep 21, 2011

### shebbbbo

The question asks to show using the residue theorem that

$\int$ cos(x)/(x2+1)2 dx = $\pi$/e
(the terminals of the integral are -$\infty$ to $\infty$ but i didnt know the code to write that)

I found the singularities at -i and +i

so i think we can then say

$\int$cos (z) / (z+i)2(z-i)2 dz

and if we take that integral over a closed contour then we will be left with the residue

and i know that once we have found the residue you can multiply this by 2$\pi$i and sum all the residues together.

BUT...

i dont know how to find the residues for this question. the cos on the top line is causing me trouble. maybe i need to expand the cos as a series? but im not sure

thanks

2. Sep 21, 2011

### Hootenanny

Staff Emeritus
3. Sep 21, 2011

### shebbbbo

ive looked on that page and stared at it for ages...

will both methods work? and am i correct with what i have done so far? (ie poles at +i and -i)???

4. Sep 21, 2011

### Hootenanny

Staff Emeritus
Yes, both methods will work and you are indeed correct in what you have done so far.