Calculus 2: The Integral of 1/x and a Rant Against Absolute Values

Longtime fan of your channel, and this made me smile: I've *always* felt the same way about this! Thank you for finally saying what needs to be said. :)

[DenysDutykh]

Unfortunately, the point of view presented in this video is also false in a similar way to what is published in standard calculus texts. The correct answer is (sorry for the absence of LaTeX on YouTube comments):

AntiDerivative(1/x) = ln(x) + C_1, if x > 0 and ln(-x) + C_2 if x < 0.

The computation of an antiderivative is essentially a topological operation and one has to watch out the singularities of the function we deal with.

And yes, the usage of terms such as primitive, integral, etc. in this context is not appropriate. The word 'integral' refers, for example, to Riemann, Lebesgue or Denjoy integrals. The process of finding the inverse to the derivative (i.e. the anti-derivative) has nothing to do with taking the limit of integral sums. However, they are connected, of course, by Leibniz rule. But this connection has yet to be established. It is not granted. For example, put a singularity between the point x_0 and x in Leibniz rule to see the issue...

Sorry if my comment may appear to be rude. I didn't want to offend anybody. I apologize in advance.

[mikexu]

Patreon, thank you so much for sharing your videos, they are treasures to me. Your comments on removing absolute symbol in lnx and forgetting about sec function are remarkable to me. I wished I had a teacher like you when I was young. My teachers always added traps in home works and exams, doing math just like walking in a minefield, sooner or later you will step on a mine and got killed. I developed math phobia because of this, but it also intrigued me in how to learn math. By temporary ignore special conditions, it allows one to see big pictures
and beautiful of math.

Totally agree to memorize less in math. After decades of learning and unlearning math, I found it is not good try to memory formulae in math, because formula is precise, if missed a tiny thing, it is wrong, while memorize other events, they may not need to be precise, for example, you colleague ask you for recap a meeting he/she missed, if you just remember main topics/conclusions, it will be good enough, so memorize other things may be useful but not math formula. There are so many formulas in each math subject, it is impossible to remember them all. I found my brain went to blank when preparing math finals, I could not memorize any formula, at the moment, I was a total idiot, then I said to myself what an idiot should know after studying the subject for this long, then knowledge gradually appears in my brain. In the process, I often got into a mediation status where I totally forgetting about time, where I was at etc, it was so exciting afterwards, so I usually brought a poem book, a hand writing calligraphy book with me too, because read them I had the same excitement.

Another wonderful aspect in your videos, you provide some history context of the subject, which is significant. I am fascinated by how math is taught to young kids, think about how long it took for human to develop numbers, decimal point, and how long it took for our ancestors to accept negative numbers, irrational number, yet each of this concept maybe
just explained to 10-12 years kids in a 40 minutes or less class!

[angelmendez-rivera]

I completely disagree. This notion of "let's focus on integration" may seem good on paper, but it turns out to be horrible pedagogy in practice. Teaching them the wrong foundations only makes it so that they follow them blindly rather than being strategic about how they integrate. And it also makes it so that they won't be able to understand the material from courses that follow immediately after calculus, unless they unlearn the foundations and relearn them, which is extremely hard and unnecessary effort, and the conflict only makes their understanding of integrals blurrier than you would think.

Students need to be taught to be careful about donains even before calculus. It's extremely important: more important than knowing how to integrate, in fact.

[alojzybabel]

Those ugly details might be ugly, but if you're not careful about them right away, you may end up with some nonsensical formula pretty quick, e.g. that -1 = 1, or a lot of similar absurdities. So I'd rather be careful now than sorry later :q

[musclesmalone]

That's hilarious. Lets have some downvotes for absolute value functions! :)