If two definite integrals are equal, does there exist a chain of substitutions and/or partial integrations which will get us from one to the other?

Earlier today I was having a little fun with Catalan's constant and its various integral representations: showing that they all do indeed evaluate to the same thing. This got me wondering whether this is always possible, if we are given several integral representations of the same real number. I then thought of a counter-example:

$$\int_{-\infty}^{\infty}e^{-x^2}\;dx=\int_0^{\sqrt{\pi}}dx$$

But I partly put this down to the fact that the integral on the left is non-elementary, whereas the one on the right is not.

What I am more interested in, is: if we consider two elementary integrals such that:

$$\int_a^b f(x)\;dx=\int_c^d g(x)\;dx$$ Does there exist a chain of manipulations which will lead us from one to the other?

One could also ask the same question about non-elementary integrals (edit: it was recently pointed out to me something like this might be a counter-example to this second case).

Solution 1:

Consider the integrals $$ I=\int_a^b f(x)\;\mathrm dx\qquad J=\int_c^d g(y)\;\mathrm dy $$ and assume that $f\gt0$ everywhere and that $I=J$. Then the following change of variable transforms $I$ into $J$. Consider the primitives $F$ and $G$ of $f$ and $g$ defined by $$ F(x)=\int_a^x f(t)\;\mathrm dt\qquad G(y)=\int_c^y g(s)\;\mathrm ds $$ and the change of variables $x=u(y)$ defined on $(c,d)$ by $$ u(y)=F^{-1}(G(y)) $$ Then $F'(u)\mathrm du=G'(y)\mathrm dy$, that is, $f(u(y))u'(y)=g(y)$, thus, this change of variables transforms $I$ into $J$. If $f$ is not positive everywhere, consider $f_c=f+c$ with $c$ large enough and apply the above to $f_c$ and to the corresponding $g_c$.

Is the change of variable $u$ admissible? This depends on your definition of "admissible" but if the functions $f$ and $g$ are elementary and if the property of being elementary is preserved by taking a primitive and by taking the inverse, then the change of variable $u$ is indeed elementary.

Solution 2:

To make this a reasonable question you need to impose some more constraints. A good constrained version of this problem is to consider only algebraic functions and integrate them over domains which are given by polynomial inequalities, where all of the coefficients are algebraic. The values of such integrals are called periods. A simple example is $\pi$ because you can get it from integrating $3/4$ over the region $x^2+y^2+z^2<1$ in $\mathbb{R}^3$.

There is a conjecture by Kontsevich and Zagier that there is an algorithm that can convert one of these integrals into another in a finite number of elementary steps, just like you ask for. However, it is just a conjecture at this point. You might enjoy reading what they have to say on this subject: Periods.

Solution 3:

Maybe consult the material on "decreasing rearrangement" of integrals. Say, in Hardy, Littlewood, Polya, INEQUALITIES.