Inside Interesting Integrals

by Paul J. Nahin

What’s the point of calculating definite integrals since you can’t possibly do them all?.What makes doing the specific integrals in this book of value aren’t the specific answers we’ll obtain, but rather the methods we’ll use in obtaining those answers; methods you can use for evaluating the integrals you will encounter in the future.This book is written in a light-hearted manner for students who have completed the first year of college or high school AP calculus and have just a bit of exposure to the concept of a differential equation. Every result is fully derived. If you are fascinated by definite integrals, then this is a book for you.

Genres: Mathematics, Physics, Science, Calculus, Nonfiction, Reference, Textbooks

436 pages, Paperback

First published August 27, 2014

About the author

Paul J. Nahin is professor emeritus of electrical engineering at the University of New Hampshire and the author of many best-selling popular math books, including The Logician and the Engineer and Will You Be Alive 10 Years from Now? (both Princeton).

Reviews

[Arav Agarwal · Rating 3 out of 5]

Overall, Interesting Integrals is a book I want to recommend, yet it falls short of being that useful in terms of teaching the few tricks it possesses. In chapters 1-4 of the book, the book performs pretty well, delivering, albeit easy, problems which really drive home the essence of the tricks and demonstrate their uses well. The introduction on Feynman’s trick, in particular, really drives home how well the author can deliver on making a trick both easy and usable. However, afterwards the book strives to get ahead of itself without a decent understanding of the level at which it is asking some readers to go for it. While the occasional step missing is not too big of a draw, the book fails at presenting the solutions in such a way to aid the reader’s learning. There were a few times where the book managed to simply, through some weird notation, become hard to decode without the assistance of Wolfram Alpha to pick up the slack. However, such practices pale in comparison to the text’s problem at providing clear solutions in the dedicated solution section of the book. After chapter 5, I honestly stopped working through the problems since half of the time I could not even get to see the solution to know if I was on the right track or if I was headed in a way contrary to my hope of solution. While I do understand that these are “challenges”, and thus should be attempted in an untimely manner, the challenge of some of these problems varies to the point where I would like to know if I used an hour of time to get to something unsolvable. Lastly, the book’s latter half tries to squeeze a lot of different tricks in the latter half all at once. Without a proper subdivision between tricks and application, it became hard to see what the point of each part was, making intuition increase rather limited unfortunately. It’s why I ended up skipping the penultimate and ultimate chapters altogether, as I feared that his method of explanation would simply cloud my intuition when I try to get a better understanding of contour integration. Altogether, this book seems unfinished and the problems are really weirdly designed. The main draws from it are the following:

1. The old tricks really work. U-subs and integration by parts can really do miracles for a problem, so long as you remember the bounds and change them accordingly.
2. Feynman’s trick is a pretty fun way to waste an afternoon and a great way to integrate stuff. It’s a good jumping off point for me too as I need to really study Calc III this year.
3. The gamma and beta functions. These definitions are some of the cooler ones and really demonstrate some of the cool new functions integration kinda brings to the table.
4. Integration can be done with power series on a higher level, and that power series can be figured out algebraically through taking the integral of a derivative.
5. Integration can also lead to diffeqs, and thus the subjects are closer than I imagined. This gets even better when you realize that such equations can also use recursion in a cool manner.
6. That I need to learn more Calc 3 and contour integration in particular.
7. That I need a good book on Fourier stuff to really understand it.
8. Never fully trust a math book that is this unpolished, and that its okay to leave a book till I have some more experience and / or find a better on.

[Stephen · Rating 4 out of 5]

Not an easy read but a very good book. Cheers, Salud, and Sköl Paul Nahin!

[Dmitri · Rating 5 out of 5]

Sometimes you want a math book for the pure enjoyment of it. Or maybe to compensate for the fact that university-level math courses are always so formulaic and boring - you simply memorize a few key approaches to, say, integration, instead of getting engaged in really creative approaches.

Of course, it's not really fair to position this as an academic text: after all, most of us would compute integrals through Mathematica, Maple and MATLAB, but CAS systems have their limits, and having a book that exemplifies some of the pathfinding that happens as a tricky integral is evaluated is, in my opinion, a great source of inspiration for anyone looking for original, creative solutions to some of the most difficult problems.

III is written in a friendly, slightly whimsical style, and is a lot of fun to read. Of course, it helps if you like integrals (and math in general).

[Bethany Epstein · Rating 5 out of 5]

very useful introduction to the subtle art of integration. you will learn not only the standard methods from calculus class but a whole slew of other tricks that will let you tame monsters.