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:
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 : )
(As of 1/26/2020, 11h25 … )
I must admit that there is one issue with the type of plots which I included in the documents linked to above:
- I plotted some indefinite integrals, in a way that compares them.
The problem with that is the fact that any indefinite integral possesses an arbitrary constant, which will displace its plot vertically in this case, by an unknown additive value to (Y). I have reassured myself that I’m not the only person who has committed this act, but do know that the practice is not 100% as it should be. Instead, what my plots should really be doing is, to plot definite integrals, the starting-point of which happens where (X=1) in the plot. However, doing it this way involves a parameter substitution, in which the parameter of the integral, let’s say (T), is not the same variable, as (X in the plot). The definite integral would then start at (T=1), and end at (T=X). Also, this would force two integrals to be computed, where only one needed to be computed before (the additional integral being at T=1), resulting in a trivial, subtracted value, which in fact, I just inserted as the arbitrary constant of the indefinite integral.
My target audience with this plot is not, the Math Expert. And so, one main fear of mine was, that other readers might get distracted by the existence of too many integrals, parameters and calculations, and so I plotted some indefinite integrals.
What I could have done would have been, to state that ‘G1(t)’ and ‘G2(t)’ are my indefinite integrals, with arbitrary constants of zero, which is already how I defined those functions. But then instead, I could have decided that ‘H1(x)’ and ‘H2(x)’ were to be my definite integrals, that are derived from ‘G1′ and ‘G2′. And in that situation, the arbitrary constant of each can be neglected, because ‘
C-C == 0‘. And so, the block of code, that defines what I wanted my software to plot, would have read:
H1(x) = G1(x) - G1(1) H2(x) = G2(x) - G2(1)
But honestly, given the reality of people who did not study Calculus, I’d also need to explain why I’d be doing so.
(Update 2/26/2020, 20h20 : )