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.

computingintegrals while the body focuses onunderstandingthe integrals. Are you really interested in whether these geometric insights lead tocomputationalimprovements? Or are you just interested in which classical integrals have a geometric interpretation? $\endgroup$ – Timothy Chow May 24 '19 at 15:598more comments