When a calculus student ends up solving this integral using partial fraction decomposition (PFD) or integration by parts (IBP), what they have actually done is successfully applied a classical algorithm. It’s like solving the Rubik’s cube by following an algorithm you found on the Internet: the heavy lifting of inventing PFD and IBP was done centuries ago by Bernoulli, Leibniz, and Gregory.

That said, the USM is, as far as I know, the only published general scheme that, for this kind of radical–rational integrals, systematically reduces the integrand to polynomial-type functions (Laurent polynomials, to be more precise). There’s no need to set up complicated PFDs or wrestle with sec³, which can be tricky with IBP. That’s what I call solving a problem, if you’ll allow me a bit of self-promotion.

You can find a first draft (I’ll soon upload a second version with benchmarks and some improvements) of the USM method on arXiv: https://arxiv.org/abs/2505.03754

#calculus #math #symmetrymatters #halfangleapproach #euler #integration

Hey, I just generalized the 𝐥𝐚𝐰 𝐨𝐟 𝐜𝐨𝐭𝐚𝐧𝐠𝐞𝐧𝐭𝐬! Take a look at it on my blog: https://geometriadominicana.blogspot.com/2025/08/generalization-of-law-of-cotangents.html

#math #geometry #trigonometry #halfangleapproach #symmetrymatters

USM really does slash the rote burden, chiefly because one handful of exponential/hyperbolic identities replaces a patchwork of separate trig, inverse-trig, radical and Euler recipes.

Article (draft): https://arxiv.org/abs/2505.03754

#math #calculus #integral #new #euler #arxiv #halfangleapproach #symmetrymatters

The first and most complete unification of classical integration techniques ever! Say hello to USM.

Draft article: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk

#math #calculus #halfangleapproach #symmetrymatters

The USM, the Dominican method 🇩🇴, has relegated Euler substitutions to mere historical relics. Modern integration has a new name.

Method draft: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk
#math #calculus #integrals #method #euler #halfangleapproach #symmetrymatters

The USM is superior to traditional methods!

Using the USM’s IV transformation formula, from the summary at the link below, you convert integrals of this type into polynomial-type integrals. The traditional method reduces it to the integral of csc³, which would require complicated reduction formulas or integration by parts. The third Euler substitution reduces it to the integral of √(t² + a²), which also requires memorizing a standard formula or integrating it from scratch. None of this is as simple as integrating a three-term polynomial-type function, which is taught in the first few weeks of any integral calculus course.

Check out the technique here: http://geometriadominicana.blogspot.com/2024/03/integration-using-some-euler-like.html
#math #calculus #integration #USM #newmethod #halfangleapproach #symmetrymatters

I gave this integral as an answer at MathSE.

\[\int_{0}^{1}\left(\frac{5}{2} \left((x - \sqrt{x^2 - 1})^{2i} + x^4\right)-1\right)\,dx=e^\pi.\]

The integral yields \(e^{\pi}\). I don't know why the LaTeX isn't displaying properly.

Link: https://math.stackexchange.com/questions/122693/is-there-a-definite-integral-that-yields-e-pi-or-e-pi-in-a-non-trivial/4861138?fbclid=IwAR3CPXtBoNDY3yc5a1HDAgfWN7QFAO2Ph91uvH7MUgSxSRjsq-t8TWYHWdY#4861138

#math #integral #eulernumber #halfangleapproach #symmetrymatters #calculus

In my latest blog post, I have derived what appears to be a new formula for solving quadratic equations. Were you aware of it?

Link: https://geometriadominicana.blogspot.com/2024/01/trigonometric-formula-for-solving.html

#math #halfangleapproach #symmetrymatters