Modern integral calculus, as we know it in the modern world, traces its origins back to the late 1600s. Having had several hundred years, mathematicians have made a lot of progress in solving problems using integration. Their work now spans thousands of integral equations, filling hundreds of pages in books dedicated to the topic.
That's why we found a recent local news article so amazing, because a Gainsville, Florida high school student has made a contribution to that canon. In the following excerpt, he describes how his inspiration came about:
Glenn Bruda came up with the idea for his calculus integration formula late one night in May 2021.
He recalled going to bed at 11 p.m. after doing some math for fun. Bruda woke up at 2 a.m. feeling parched. He grabbing a glass of water and walked back to his bedroom when the new math formula appeared in his head.
"And I jotted it down really quickly on a notebook. And then I went to sleep," Bruda said.
The formula was a new integration technique that can be used to solve a calculus equation. The name of the technique: the Maclaurin Integration.
Bruda has described it this way: "Maclaurin Integration is a new series-based technique for solving infamously difficult integrals in terms of elementary functions. It has fairly liberal conditions for sound use, making it one of the most versatile integration techniques."
That's not something that happens every day, so we went to his pre-print paper to learn more. In it, he gives two examples of equations that do not have integral solutions, which we've presented on a single line below:
It is possible to integrate these equations to find solutions, but it takes a remarkable amount of work to do it, often using advanced techniques using Sine Integrals, Si(x), or probability based Error function Integration, Ei(x).
Or, if you want to save time over these methods, you could use a famous integration trick introduced by Richard Feynman. The following video by blackpenredpen shows how to use Feynman's technique to integrate Bruda's first example, sin(x)/x, from 0 to infinity:
Having seen that, let's look at why the Maclaurin series might provide a much easier path to find integral solutions. The next video, by patrickJMT, uses a Maclaurin series to find an approximate solution for another difficult equation to integrate to a desired level of accuracy:
This particular video does an excellent job in explaining the steps in the math being done as the solution is developed. Which we think makes the third video, by The Elements of Math, in which the Maclaurin series is specifically used to find an approximate solution to the definite integral for sin(x)/x, a lot easier to follow:
This third video was published less than a month before Glenn Bruda's middle-of-the-night inspiration. That timing raises a question. If other mathematicians are already using Maclaurin series to deal with difficult to integrate equations, what has Bruda contributed to the world's mathematical knowledge?
What Bruda brings to the table that's new is a standardized and simplified algorithm for setting up the Maclaurin series math to be able to perform the various integrations for which integral solutions are not able to be expressed using elemental functions. The method he introduces makes the computation of their solutions easier to execute that can be applied to a wider array of problems than existing methods. Anyone who has ever had to find a solution for a mathematical equation that does not have a direct solution using numerical methods can appreciate that achievement. We'd love to see Bruda further develop his pre-print paper by providing the solutions for each of the examples he presents in an appendix, if not in the main body of the paper itself.
All in all, Bruda's paper represents the kind of hands-on, practical application of maths we like to celebrate! Which is why we are.
References
Bruda, Glenn. Maclaurin Integration: A Weapon Against Infamous Integrals. Arxiv. 30 January 2022. DOI: 10.48550/arXiv.2201.12717. [PDF Document].
Labels: math
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