[The following is based on a William Lowell Putnam Mathematical Competition problem.]

Consider the definite integral:

$I = \int\limits_2^4 \frac{\sqrt{\log (9-x)}}{\sqrt{\log (9-x)}+\sqrt{\log (x+3)}} \, dx$

Mathematica's numerical integration yields the answer directly:

NIntegrate[Sqrt[Log[9 - x]]/(Sqrt[Log[9 - x]] + Sqrt[Log[x + 3]]), {x, 2, 4}]

(* 1. *)

However, the symbolic integration does not:

Integrate[Sqrt[Log[9 - x]]/(Sqrt[Log[9 - x]] + Sqrt[Log[x + 3]]), {x, 2, 4}]

Note, though, that with the proper substitution, the integral can be solved symbolically. As the integration variable $x$ goes from $2 \rightarrow 4$, then the term $9 - x$ goes $7 \rightarrow 5$ and the term $x + 3$ goes $5 \rightarrow 7$. This symmetry suggests the substitution $x = 6 - y$ and reversing the limits of integration. This substitution gives the integral:

$I = \int\limits_2^4 {\sqrt{\log (y+3)} \over \sqrt{\log (y+3)} + \sqrt{\log (9-y)}} dy$

and so:

$2I = \int\limits_2^4 {\sqrt{\log (x+3)} + \sqrt{\log (9-x)} \over \sqrt{\log (x + 3)} + \sqrt{\log (9 - x)}} dx = \int\limits_2^4 dx = 2$

and thus $I = 1$.

Mathematica would have to be quite "clever" in finding the symmetry of the integrand given the integral's limits and performing the substitutions.


Is there any general approach (with FullSimplify, for instance) in which Mathematica could solve this integral symbolically?


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