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This question came up when I was reading through this question.

Are there definite integrals which cannot be computed using any real analysis techniques but are amenable using only complex analysis techniques?

If not, is there any reason to believe that if a definite integral can be evaluated using a complex analysis technique, then there must exist a way to compute the same definite integral using only real analysis techniques?


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    An opposite question may be of interest http://math.stackexchange.com/questions/126655/when-a-real-valued-integral-cant-be-evaluated-using-contour-integration2012-07-06

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Yes, there are so many such definite integrals that cannot be solved by real analysis techniques, such as

$\int^{\infty}_{0}\frac{dx}{1+x^n}.$

By the Cauchy's integral formula, we can compute it simply.

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    This integral can be evaluated with the beta function. We don't need complex analysis.2018-07-11