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When I started to learn mathematics, I was fascinating by legendary «Демидович»: problems in mathematical analysis. Fifteen years later, when I open chapters about integrals, I see a long list of periods of mixed motives and generalizations of those. In particular, for almost each integral I can try to construct an appropriate algebraic variety, look at the weights on cohomology and so on. Is all this information helpful in understanding these integrals? I think yes. Here are three examples.

Abelian integrals

We fix a polynomial relation $P(x,y)=0$ and consider integrals $$ \int_a^b R(x,y)dx, $$ where $R$ is a rational function. Here the theory of Abelian functions suggests that the genus of the normalization of the curve $P(x,y)=0$ is the key invariant. For instance, if $g=0$ it leads to Ostrogradsky method and Euler substitutions, described in Demidovich’s book.

Mixed Tate Motives unramified over $\mathbb{Z}$ When I see an integral like $$ \int_{0\leq x\leq y\leq 1} \frac{dx}{1-x} \frac{dy}{y} $$ I can interpret it as a period of a cohomology group $$ H^2(\overline{M_{0, 5}}-A, B-A \cap B)) \quad. $$ This group carries mixed Hodge structure of mixed Tate type, so we are dealing with mixed Tate motive. One can check that it is unramified over $\mathbb{Z}$. A theorem of Francis Brown implies that the integral above is a rational linear combination of multiple zeta values.

More general mixed Tate motives

This is conjectural. Consider an integral, defining a period of mixed Tate type. In view of Goncharov conjectures this integral should be a rational linear combination of multiple polylogarithms. This implies in particular that volumes of hyperbolic polytopes in all dimensions could be expressed via multiple polylogarithms.

Question: I would like to know about other general approaches to computing integrals and see more explicit examples of those. Here by “computing” I mean restricting significantly the set of functions, containing the answer, like in the examples above.

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    $\begingroup$ If one restricts to some class of integrals, there is a hope to express all integral in the class via a relatively small subset of those viewed as "special functions". In these case the numbers you obtain are far from being random. You can look at the paper by Kontsevich and Zagier (maths.ed.ac.uk/~v1ranick/papers/kontzagi.pdf) for a general setup of motivic periods. $\endgroup$ – Daniil Rudenko May 24 '19 at 13:44
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    $\begingroup$ OK, I see. I think in the motivic science you restrict to rational functions or something like that. Maybe you could phrase your question in a different way, something like "for which classes of functions there is a more conceptual point of view on integrals?". Note that at the moment this comment is posted, the question says "general approaches to understanding integrals", which I personally find misleading. $\endgroup$ – user140765 May 24 '19 at 13:46
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    $\begingroup$ Actually, there is a more general notion of exponential periods. Hypergeometric functions, Bessel functions and so on also could be analyzed using cohomology with coefficients in a local system. So almost any integral I have ever seen in a textbook fits in some natural class like that. $\endgroup$ – Daniil Rudenko May 24 '19 at 13:48
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    $\begingroup$ @DaniilRudenko : The title of the question does not seem to match the question in the main body. The title mentions computing integrals while the body focuses on understanding the integrals. Are you really interested in whether these geometric insights lead to computational improvements? Or are you just interested in which classical integrals have a geometric interpretation? $\endgroup$ – Timothy Chow May 24 '19 at 15:59
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    $\begingroup$ For functions with elementary integrals, there is Risch algorithm: en.wikipedia.org/wiki/Risch_algorithm $\endgroup$ – Margaret Friedland May 25 '19 at 0:54

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