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 Here 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 understanding integrals and see more explicit examples of those.