1. The problem statement, all variables and given/known data I want to find the value of the integral: ∫cos(x)/((x+a)2+1) dx from ]-∞;∞[ 2. Relevant equations Residue theorem 3. The attempt at a solution My question is seeking more a conceptual understanding of why transforming to the complex plane works. According to Jordans Lemma the semi arc of your contour will go to zero if the maximum of lf(z)l -> 0 as lzl->∞. For the function f(z)= cos(z)/((z+a)2+1) - how can you check that this holds? If Jordans Lemma isn't satisfied is it still possible to calculate the real integral by transforming to the complex plane, calculating the integral, and subtracting the contribution from the semi arc?