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I am trying to learn contour integration, for the purpose of learning another technique I can apply to computing definite (real) integrals.

Although I can apply the process well enough, ( with some issues in contour selection, but that's a different question). The problem i am dealing with, is how can a person tell if a particular real integrand is a good candidate for contour integration?

I've read something about "five cases" where it can be used but it felt weird for such an arcane technique to be limited to precisely five situations so I came here to ask instead.

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    $\begingroup$ I really think this is a good question. It feels like passing real integrals to residues is a magic trick. $\endgroup$ May 3, 2023 at 2:03
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    $\begingroup$ This, this, this, and this are non-intuitive examples of contour integration I made up. $\endgroup$ May 3, 2023 at 4:59
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    $\begingroup$ This could also give you some ideas. I'm not sure what you mean by "five cases," but you're generally free to apply contour integration to any (in)definite integral you want as long as divergence doesn't play a role. In my naive experience, I've noticed that integrals with a period of some multiple of $\pi$ and improper integrals over $\mathbb{R}$, $\mathbb{R}^-$, and $\mathbb{R}^+$ with rational polynomials/logarithms/exponentials tend to be candidates. $\endgroup$ May 3, 2023 at 5:22

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