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I'm aware that some "real integrals" can be evaluated using complex integration, making some problems which would be extremely difficult by real methods much easier. Is it possible to evaluate all "real integrals" that could be evaluated by complex integration, by only real methods?

It would be fair to say that I'm a novice (if that) in this area, hence, by "real integral" I mean an integral of the form; $\int_a^b f(x)dx$, where $a,b \in \mathbb{R}$ and the domain and range of $f(x)$ are restricted to real numbers.

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    $\begingroup$ I recommend "Inside Interesting Integrals" by Nahin. Although I do not believe every integral can be done without contour integration, there are some devilishly clever method to do integrals without any complex analysis. $\endgroup$ – Rellek Jan 9 '18 at 0:39
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    $\begingroup$ It does seem to be the case that a small "detour" through the complex plane often simplifies the process of evaluating many integrals. That said, one integral I would love to see evaluated using real methods only is the integral $$\int_{-\infty}^\infty \frac{dx}{(e^x - x)^2 + \pi^2} = \frac{1}{1 + \Omega},$$ where $\Omega$ is the omega constant. $\endgroup$ – omegadot Jan 9 '18 at 1:10

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