Post by: paul cotter on 27/04/2022 12:19:27
Post by: Eternal Student on 27/04/2022 13:58:08
First thing: Ln(x) is a function that has a singularity when x=0. To say that another way, Ln (0) = undefined.
That is the first hint that the indefinite integral will be complicated.
I just used an online integral calculator, this one: https://www.integral-calculator.com/ but there are many others you could use.
Anyway that online calculator will show you the steps involved in performing the integral if you're interested.
Here's their solution:
integral.JPG (61.92 kB . 847x720 - viewed 2160 times)You'll notice it makes use of some special functions like Li and Ei. These are explained in Wikipedia (or other places):
https://en.wikipedia.org/wiki/Logarithmic_integral_function
https://en.wikipedia.org/wiki/Exponential_integral
These are not elementary functions - but we were expecting trouble right from the start because we knew that Ln(x) was singular somewhere. There may not be any elementary function that will serve as the anti-derivative (LATE EDITING: I'm now fairly sure that there isn't). You also have to be aware that you have a solution which is only valid on part of the Real Numbers (i.e. not everywhere).
For definite integration, i.e. integration between upper and lower integral values, provided you don't integrate over a range where the function(s) become singular you will be perfectly able to get a numerical value with that solution.
Best Wishes.
Post by: paul cotter on 27/04/2022 17:53:28
Post by: Eternal Student on 27/04/2022 20:49:03
I used hyperbolic substitution which led to a horrid degree of expansion leading to several pages of calculations.I'm sure there are many ways of finding the integral. The modern mathematics software like Mathematica and the various on-line tools (like the one mentioned earlier) have basically devalued that skill. I'm old enough to think that's a bit of shame but I suppose we've got to look on the positive side of things: What I should be saying is that the Mathematical software is amazing these days.
I do understand that some simple expressions cannot be integrated directly. what say you?That is actually a big and complicated topic. I don't suppose you want to talk about Lebesgue integration - because there's only 3 people in the world that do and they have a forum all of their own - so I'll just skip to the conclusion:
Provided a function, f, satisfies some sensible criteria (for example that is a measurable and bounded function), then there will always be a function, F, that is its anti-derivative (or indefinite integral if you prefer that term). It's just that sometimes the simplest form to express that antiderivative is just to say that is equal to itself, there's no simple expression for it as an elementary function (a combination of conventional functions you would have seen before like polynomials and trig. functions etc.).
Best Wishes.
Post by: paul cotter on 28/04/2022 10:29:07