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).
Best Answer
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.