Ideas from LLMs (sad, but still interesting).
Today’s suspect
The integral that we will be looking at is Daily Integral’s 9th Oct 2026 Beginner Integral
When I first saw this, a thought went through my mind: “integration by parts in beginner?” Things seem to be getting stale in the beginner section, and yet, IBP was very easy to spot in this case. So, instead I wondered what other creative ways are there to solve this seemingly benign integral.
Feynman?
Those who are familiar with “Feynman integration” or “Differentiation under the integral sign” may notice we have the factor poking out, begging for Feynman’s trick to be used. So first, we parameterize the 2:
Then differentiate our new function with respect to α:
We also have another way to write :
Putting it all together:
Double trouble
If you recall, natural logarithm can be defined by an integral:
So we will substitute that into our integral!
Now, let’s swap the order of the integral. We have , so the integral becomes
Going exterior
The previous method relied on calculating the interior weighted area of a triangle. But with Green’s theorem —a special case of Generalized Stokes theorem, I just have to say it :)—we can compute path integrals instead
Choosing (and renaming to ) gives us
Where is the triangle with vertices . Analyzing the edges, and their directions we get
Differential algebra
Though I admit I don’t know much about this field, the suggested proof seems easy enough to understand, at least on the surface level. So, I will be sharing it here.
Consider the differential field equipped with derivation . The original integrand lies in the subspace . Let be a “guess” or (ansatz) for ; it should have the form
This gives us
Note that a has to be a polynomial, meaning the additive constant of integration of has to be zero. Therefore, , and
Since , by the fundamental theorem of calculus,
Conclusion
Integration is, itself, an art, the art that I still find difficult to truly wield. There are still plenty of unconventional ways to integrate . Sadly however, I do not yet have the knowledge required to fully grasp them. Maybe another time!