integration - Evaluating the complex integral $int_{-infty}^infty frac{cos(x)}{x+i},dx$

I stumbled upon this particular integral a few minutes ago, and I have no idea how to go about it :

$$\int_{-\infty}^\infty \frac{\cos(x)}{x+i}\,dx$$

I looked up on the internet and I managed to find out that something called residue should be taken into account when dealing with such integrands, but I'm clueless at this point in matters of complex analysis. As context, a colleague of mine suggested this could be an interesting exercise.

Any ideas ?

Answer

If you are not aware of the residue theorem, things are harder, but not impossible.

Step 1. The integral is converging in virtue of Dirichlet's test, since $\cos x$ has a bounded primitive and $\left|\frac{1}{x+i}\right|$ decreases to zero as $|x|\to +\infty;$

Step 2. By symmetry (the cosine function is even) we have: $$I = \int_{\mathbb{R}}\frac{\cos x}{x+i}\,dx = -2i\int_{0}^{+\infty}\frac{\cos x}{1+x^2}\,dx$$ so we just need to compute a real integral;

Step 3. We may compute the Fourier cosine transform of $e^{-|x|}$ through intergration by parts. That gives that the CF of the Laplace distribution, by Fourier inversion, is everything we need to be able to state: $$\int_{0}^{+\infty}\frac{\cos x}{1+x^2}\,dx = \frac{\pi}{2e}.$$