Find $\int \frac{\ln(1-x)}{x}dx$

Question

I would like to find $$\int \frac{\ln(1-x)}{x}dx$$

On the site where I found this the problem was actually slightly different; the question was to evaluate $$\int_0^1 \frac{\ln(1-x)}{x}dx$$ which can be solved using its Maclaurin series expansion and the solution for the Basel problem, but I was wondering if there's a way to directly find the antiderivative of the integrand. I have tried substitutions included hyperbolic and polynomial ones, such as $x=\sinh^2 x$, but I haven't managed to get anywhere with it.

Would you mind giving me a push in the right direction?

Thank you for your help.

Maybe this may help you: https://www.wolframalpha.com/input/?i=integrate+ln%281-x%29%2Fx — user0102, Dec 21 '20 at 22:17
@APCorreia thank you, so there's no solution in 'nice' functions then? — A-Level Student, Dec 21 '20 at 22:18
Have a look at this https://en.wikipedia.org/wiki/Spence%27s_function — Donald Splutterwit, Dec 21 '20 at 22:19
It's definitely not boring! I think it's quite interesting that even though we cannot find an anti-derivative - we can still find the solution to the definite integral. The large majority of integrals in fact do not have nice functions as solutions! — Riemann'sPointyNose, Dec 21 '20 at 22:25
@Riemann'sPointyNose but aren't we really just inventing new functions to solve a problem? That seems like cheating. — A-Level Student, Dec 21 '20 at 22:27
@A-LevelStudent a couple of things to this question: ${(1)}$ we aren't inventing any new functions by using the Maclaurin expansion, this is perfectly valid. We just have to justify switching the summation and the integral (which is fine, we can do it here) and ${(2)}$ no! It isn't cheating. Maybe to put it another way - imagine we only ever knew about polynomials. Then what is ${\int \frac{1}{x}dx}$? It can't be written in terms of a polynomial. Does that make it cheating to define ${\log(x)}$? Of course not. We only ever deal with a very small, pathetic list of functions that actually exist! — Riemann'sPointyNose, Dec 21 '20 at 22:30
@A-LevelStudent I guess really all Mathematics is about is taking previous work and extending it. So defining new functions definitely isn't cheating, the fact is that's what we do all the time — Riemann'sPointyNose, Dec 21 '20 at 22:31
@Riemann'sPointyNose thanks for explaining this to me, this point has been bothering me for a while and you have helped, especially with your point $(2)$. Just to clarify, as I wasn't clear; I didn't find using the expansion cheating, I found creating Spence's function cheating. — A-Level Student, Dec 21 '20 at 22:34
@Riemann'sPointyNose Maybe the reason I find functions similar to Spence's one annoying is because I can see no other definition of them outside of the integral, whereas with trig, polynmoial, exponential and logarithmic functions they can be defined without integrals. — A-Level Student, Dec 21 '20 at 22:36
@A-LevelStudent no problem! And ah sorry I didn't realise this. But yeah, hopefully this argument still helps out! And yeah I understand that, there is some sort of seeming difference between say, spencers function and log. But it's still completely valid to define a function in terms of an integral :) at the end of the day, it's just a symbol that we are referring to — Riemann'sPointyNose, Dec 21 '20 at 22:37
@Riemann'sPointyNose fair enough, I suppose I'll just have to grin and bear it :) Thanks again for your help! — A-Level Student, Dec 21 '20 at 22:38

score 5 · Accepted Answer · answered Dec 22 '20 at 03:58

The Euler dilogarithm function $~Li_2(x)~$ defined for $0 ≤ x ≤ 1$ by $$Li_{2}(x)=\sum_{n\ge1}\dfrac{x^n}{n^2}=-\int_{0}^{x}\frac{\ln{(1-t)}}{t}dt$$is one of the lesser transcendental function. Nonetheless, it has many intriguing properties and has appeared in various branches of mathematics and physics.

So $$\int_0^1 \frac{\ln(1-x)}{x}dx=-Li_2(1)=-\dfrac{\pi^2}{6}$$

You may read the following links (and the references therein) for further knowledge about this type of integrals:
"Dilogarithm Identities" by Anatol N. Kirillov
Calculate a $\operatorname{Li}_{2}(-1)$ using Integral Representation
Dilogarithm by Wolfram MathWorld
The Dilogarithm Function by Don Zagier
Spence's function by Wikipedia

Find $\int \frac{\ln(1-x)}{x}dx$

1 Answers1