Integral wonder woman

By Lode Vermeulen • 2025-05-05 • 5 min read

#math#internet culture#beginner friendly

Taking integrals analytically can be attempted for arbitrarily hard functions. Most of these are beyond the reach of mere humans. The mysterious wonder girl Cleo makes it look easy.

Integrals can be seen as one of the most well contained class of mathematical problems to solve. Notationally, integrals consist of three parts. The defining symbol of an integral is the "elongated S", $\int$ . After that comes the function and the variable you are integrating over. A very simple example looks like this:

\int xdx=\frac12x^2+C \ .

That's pretty much it, all integrals will take a form that looks like this. Despite its simple description, solving integrals analytically is its own field of research. The complexity lies in the function $f$ you are integrating over (here, $f=x$ ). Let's look at a more elaborate example:

\int_{-1}^{1} \frac{1}{x \sqrt{1+x} \sqrt{1-x}} \ln \left( \frac{2x^2 + 2x + 1}{2x^2 - 2x + 1} \right) dx=4\pi \text{arccot}{\sqrt \phi} \ .

This integral, and the fact that it has such an elegant closed form solution, is particularly of note. The right hand side consists of three usually unrelated parts, the well known constant pi, the inverse cotangent function and the golden ratio. Getting to this solution requires mathematical training equivalent to an undergraduate math degree and several pages of derivations containing multiple non-trivial steps. Furthermore, this is the most upvoted solution of Cleo on the mathematical Q&A forum mathexchange.

Uncovering the mystery of Cleo

Cleo has a habit of posting simple looking solutions to impossible looking integrals. Often she posted her answer within days of the question being posted and, most notably, without showing any derivations. Not everyone enjoys these type of answers. Most users who ask for solutions of integrals are more interested in how to evaluate the integral. This way, they can use the techniques to generalize the result or to apply it to other integrals. The final numerical value of a single integral is often not too useful on its own, especially considering they can be approximated pretty well numerically. Nonetheless, she kept posting them. As another (purposefully scary looking) example, there is the result:

\int_0^\infty{_3F_2}\left(\begin{array}c\tfrac58,\tfrac58,\tfrac98\\\tfrac12,\tfrac{13}8\end{array}\middle|\ {-x}\right)^2\frac{dx}{\sqrt x} = \frac{50\,\pi^{3/2}}{3\,\Gamma^2\left(\frac14\right)}\Big(\ln\left(3+\sqrt8\right)-\sqrt2\Big) \ .

To which there were the responses:

@Aritro Shome any proof, solution?

@AlanSTACK No proof needed, I trust her implicitly. At this point, she might as well be God.

While she probably is not God, it is not immediately clear what her real identity is. After all, someone who can solve such complicated integrals so quickly must be a pretty good mathematician, right? Her profile does not shine much light on this, either. Besides apologizing for the fact that she cannot post long answers due to a medical condition, not much more information can be found. As Cleo and her solutions have still been gaining more and more attention online in recent years, even though all activity on the account was between 2013 and 2015, this prompted some people to start an investigation into her true identity.

The first most notable investigation was made by Corey, who put all known activity connected to Cleo in a timeline and did some data analysis on accounts that share similar activity times to Cleo. While that research was inconclusive, some plausible theories were listed on the mysterious Mathexchange user. A lot of possibilities could actually be ruled out. One of the most adopted theories remains that all the solutions were genuinely calculated by hand by a math genius. Another theory is that the person running the account has access to an advanced computational tool, which would also explain the constant lack of derivations.

Following a hit short, the Youtuber Joe McCann revived the investigation into Cleo.

section on the legend that Cleo became
section on the uncovering of Cleo's identity
section about how Cleo came to the answers: computer algebra systems Yet, we know that the answers are correct, as we can numerically compare the value of the integral to Cleo's solutions.
section on why Cleo chose to post like this
section on the effect of Cleo's way of posting
section on what now: say that Cleo has stopped posting for years now and that her account has been closed for the following century Due to her not following Stackexchange guidelines, the account has been banned this January until the year 2125.

links:

https://www.reddit.com/r/math/comments/1qpus4/comment/cdfysit/?context=3
https://www.reddit.com/r/HobbyDrama/comments/130miwv/math_se_cleo_and_the_angry_mathematicians/?chainedPosts=t3_z4dyzr%2Ct3_1qpus4
https://www.reddit.com/r/math/comments/1iklgmt/stackexchange_cleo_mystery_cleo_is_finally/?chainedPosts=t3_130miwv%2Ct3_z4dyzr%2Ct3_1qpus4
https://www.reddit.com/r/math/comments/1d9eks9/was_cleo_actually_a_math_stackexchange_genius/?chainedPosts=t3_1iklgmt%2Ct3_130miwv%2Ct3_z4dyzr%2Ct3_1qpus4
https://www.reddit.com/r/mathmemes/comments/1com4bu/that_story_was_too_good_to_be_true/