Oh boy! Integrals IV is now live! This one's got all the juicy integration techniques in it. All of the previous sections have been leading up to this one. This is the meat and potatoes, folks. We've got your u-substitution and your integration by parts, and we've even got your trigonometric substitution and your partial fractions. These are the methods generations of mathematicians have learned and applied time and again for both business and pleasure, time and time again.
I hope that these sections in particular can become great repositories of interesting problems over time. Many problem sets, textbooks, and Q&A sites contain all kinds of interesting examples, and I'd like to include as many of them as I can all in one place. For now, each section has at least 10 problems, but someday I'd like them to be definitive collections.
Integrals III is here! The section starts off with an overview of the Fundamental Theorem of Calculus. The theorem is simply given for now, although a proof may come at some point. Subsequent sections provide some practice problems for integrating simple functions such as polynomials, exponential functions, logarithms, and trigonometric functions. Indefinite integrals are given as formulas, once again with proofs elided. In no case is the proof left as an exercise to the reader without a solution, since that would be evil. Future sections will cover more advanced integration techniques, such as substitution, as well as applications, such as calculating arc length.
For some reason you want to learn, or are forced to learn, proof-based mathematics. Algebra clicked for you at some point, you conquered calculus, and you then went on to summit the mountains of multivariable calculus, linear algebra, and differential equations of both the ordinary and partial kind. But now you've been tasked with proving something. Something horribly simple. Something obvious. Something so obvious, in fact, that you don't even know how to prove it. It's just...true. Take, as my favorite example, the fact that the square root of 2 is irrational. Suddenly you can't help but want to hide in a hole so that nobody can see you. You were blasting through the wave equation and had all of those sexy partial differential symbols dripping off your page, but now you've been knee-capped by some kind of peasant math that was supposed to stay in middle school!
"Left as an exercise to the reader" (LAAETTR) are perhaps the most hated 7 words in all of mathematics. Every student, whether dying to just finish a problem set for a required class or genuinely trying to attain a robust understanding of a new concept, has encountered these infamous words. They represent the worst education has to offer - the tantalizing promise of understanding, swiftly obliterated. And worse, this obliteration is nominally done for the sake of the student. Rigor! The process of learning! You need to figure it out! Nonsense.
The second section on integrals has arrived! The section covers three additional methods of estimating integrals using finite sums: the midpoint method, the trapezoid rule, and Simpson's rule. The section on Simpson's rule features an exhaustive derivation of the formula, although an explanation of Lagrange interpolation is saved for later. The derivation is provided more for completeness than as an actual practice problem, as its absence would have bothered me. While knowledge of solving integrals analytically is technically required to understand how Simpson's rule works, it is nonetheless grouped with the other estimation methods in keeping with the general theme.