[paul cotter (OP)]

many(very many) years ago a fellow student asked if I could integrate a seemingly simple expression, xdx/linx. neither he nor I could do this and over the years I have tried many times without success. about five years ago I tried again and I believe I succeeded, but I could have made a mistake as my "solution" is several pages long involving hyperbolic substitution followed by integration by parts. I do understand that some simple expressions cannot be integrated directly. what say you?. just to be clear i'm talking about x times dx divided by the nat log of x.

[Eternal Student]

Hi.

   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 2159 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.

[paul cotter (OP)]

thank you, very interesting. those transcendental functions are new to me as my maths education didn't go that far. as I said I used hyperbolic substitution which led to a horrid degree of expansion leading to several pages of calculations.

[Eternal Student]

Hi.

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.

[paul cotter (OP)]

fascinating stuff, thank you very much. I agree with you that the manipulations required to solve these problems will most likely become a lost art when an online calculator does it all for one. this is an inevitable consequence of progress as many fine disciplines become redundant.