An affirmation of a concept that exists in Calculus 2, the Integral of (1/x).

There are certain concepts in Calculus 2, which introduces definite and indefinite integrals, that are taught to College and University Students, and which are actually considered to be basic information in Higher Math. One of them is, that the integral of (1/x) is the natural logarithm of (x).

Yet, some people just like to go around and dispute such things, much as the concept is popular, that (2+2) does not equal (4). And so, what I have just done is to ignore the obvious fact, that people who studied Calculus at a much higher level than I have, have found an analytical proof, and to ask the question:

‘What would happen if the integrals of simple power functions were given, that have powers slightly more-negative and slightly more-positive than (-1), in relation to this accepted answer, the natural logarithm of (x)?’ The accepted answer should always fall between those two curves, even if some plausible arbitrary constant is added to each power-function integral, such as one which sets all the functions to equal zero, when the parameter equals one. Not only that, but it’s easy for me to plot some functions. And so, the following two worksheets have resulted:

Testing the Integral of (1/x) – EPUB File for Mobile Devices

Testing the Integral of (1/x) – PDF File for Desktop and Laptop Computers

Further, I’d just like to remind the reader, that a function can easily be defined that follows a continuous line, except at one parameter-value, at which it has a different value, such that the neighbouring intervals in the domain of said function do not include this endpoint, in either case. The only question which remains is, whether that function is a correct answer to a question. And, because such functions are possible, the answer depends on additional information, to the idea that there are exceptions to how this function is to be computed.

(Update 1/26/2020, 20h20 : )

Continue reading An affirmation of a concept that exists in Calculus 2, the Integral of (1/x).

Two Examples of Improper Integrals

In a recent posting I proposed to answer a question using an indefinite integral, which would more-correctly have been solved using the corresponding, definite integral. The issue there was that if this integral was rewritten as some arbitrary definite integral, this could in some cases have resulted in what’s called an ‘improper integral’. And what my reader may not realize, is that improper integrals exist, with well-behaved solutions, just as some infinite series converge.

And so, I have written a work-sheet below, which reminds people who may not remember their Calculus 2 exactly, of what forms improper integrals can take:

Link to a Letter-Sized PDF File

Link to an EPUB File for Phones

 

(Edit 6/05/2019, 18h25 : )

I have just revised the work-sheets above, to include some plots, and to provide a clearer understanding to anybody who might be interested in them, but who did not study Calculus 2. But some readers of the EPUB version may notice wonky formatting.

When I export Math notation to regular HTML, or to anything which is based on regular HTML, such as to an EPUB File which is not using MathML, I am faced with a problem every time correct Mathematical notation requires that 3 glyphs be stacked, as is the case with the (definite) integral operator, and with the Sigma operator, the latter of which denotes a summation. The only way I see around this issue is, to give the operator in question both a subscript and a super-script.

While the result can be read and understood, doing so requires additional concentration by the reader. I’ve written earlier postings, in which I described this problem, but the advantages here are, a notation which regular EPUB readers can display, as well as my ability to include the Computer Algebra and thus the plots, of “SageMath”, using the “LyX” graphical front-end to LaTeX, which makes the typesetting easier for me.


 

This limitation does not exist when only exporting the results to a PDF-File. But, in order to take advantage of the more-correct formatting of the resulting PDF File, I’d need to create two separate versions of my own document, one for export to (exact) PDF, and one for export to (messy) EPUB. While I did this amount of work for simpler work-sheets, I’m unwilling to do this for the more-complex work-sheet I just linked to.

 

(Edit 6/05/2019, 18h55 : )

There’s an added challenge to me, in the form of something my particular software is unable to perform. When I’m using LyX to typeset my work-sheets, the following two possibilities emerge:

  • Those work-sheets may consist entirely of Math written in my own hand, in which case I am able to export them to an XHTML File that contains MathML, and this will enable me to set up a master document, in which 2 but not 3 elements can be stacked. Then, the ‘Limus’ notation will be correct, and the integrals will be so-so. But the resulting master document can then be exported in two ways that eventually end up as a PDF and as an EPUB3 File, the latter requiring MathML from the EPUB reader app.
  • Those work-sheets may contain Computer Algebra and/or Plots that are essential, in which case only .TEX Files of the SageTex variety can be exported, which in turn can only be converted into plain HTML. This will result in an EPUB File that is inferior, but that all mobile EPUB reader apps can view. But simultaneously, through a separate master document and additional work on my part, a pristine PDF File can result, which still requires a full-sized monitor or other output device to read.

So, unless I find ways to export SageTex Files specifically, to XHTML with MathML, I’ll be facing issues in how to create typeset documents in the near future.

Dirk