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Evaluate [itex]\int[/itex]dx/(x^{2}+2x+2) (limits from -∞ to ∞) by contour integration

Ok so, this is what i did...:

we have ---> [itex]\Gamma[/itex] : |z|=R

[itex]\oint[/itex]dz/(z^{2}+2z+2) =

[itex]\int[/itex]dx/x^{2}+2x+2 (limits -R to R) +[itex]\int[/itex]dz/(z^{2}+2z+2) (lower limit [itex]\Gamma[/itex])--> (1)

[where f(z)=1/(z^{2}+2z+2)]

we have, lim_{z→∞}zf(z) = lim_{z→∞}z/(z^{2}+2z+2)

= 0

therefore,

lim_{R→∞}[itex]\int[/itex]dz/(z^{2}+2z+2) [lower limit[itex]\Gamma[/itex]

= 0

Taking the limit R→∞ on both sides of (1)

[itex]\oint[/itex]dz/(z^{2}+2z+2)= [itex]\int[/itex]dx/x^{2}+2x+2 [limits from -∞ to ∞) + 0

where f(z)= 1/z^{2}+2z+2 so for finding out the poles of f(z)....

yeah this is where im stuck, cuz in the previous one we did f(z) was→1/z^{4}+1

so there we solved for z the way we solve for the roots of a complex number...

all that k= n-1 and nth root of z = re^{i(θ+2k∏)1/n}......

so how do we solve in this case?

Thanks alot in advance....

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# Homework Help: Problem with contour integration

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