# A particular vanishing integral

While dealing with a definite integral on AoPS I discovered (I have to admit by pure chance) the following relation

$$\int_0^1\log\left(\frac{(x+1)(x+2)}{x+3}\right)\frac{\mathrm dx}{1+x}~=~0\tag1$$

The proof is quite easy, but feels kind of contrived. Indeed, just apply a self-similar substitution - $$x\mapsto\frac{1-x}{1+x}$$ - to the auxiliary integral $$\mathcal I$$ given by

$$\mathcal I~=~\int_0^1\log\left(\frac{x^2+2x+3}{(x+1)(x+2)}\right)\frac{\mathrm dx}{1+x}$$

And the result follows. However, to consider precisely this integral seems highly unnatural to me (in fact, as I mentioned earlier, this integral was just a by-product while evaluating something quite different and I discovered $$(1)$$ when experimenting with various substitutions).

The crucial point to notice concerning $$\mathcal I$$ is the invariance of the polynomial $$f(x)=x^2+2x+3$$ regarding the self-similar substitution which allows us to deduce $$(1)$$. Additionally for myself I am quite surprised by the special structure of $$(1)$$ since we have factors of the form $$(x+1)$$, $$(x+2)$$ and $$(x+3)$$ combined which calls for a generalization (although I found none yet).

It there a more elementary approach, not relying on such an "accident" like examining the integral $$\mathcal I$$ for proving $$(1)$$? Additionally, can this particular pattern be further generalized? Answers to both questions (also separately) are highly appreciated!

• The $\displaystyle\log$ argument numerator -in the $\displaystyle\mathcal{I}$ definition- must be $\displaystyle x^{2} + 3x + 2$ instead of $\displaystyle x^{2} + 2x + 3$. – Felix Marin Aug 27 '20 at 3:27
• @FelixMarin No. By my calculations $\left(\frac{1-x}{1+x}\right)^2+2\left(\frac{1-x}{1+x}\right)+3=2\frac{x^2+2x+3}{(1+x)^2}$ which is what the approach is all about. But $\left(\frac{1-x}{1+x}\right)^2+3\left(\frac{1-x}{1+x}\right)+2=2\frac{x+3}{(x+1)^2}$ which yields an interesting other identity but not the one the post is concerned with. – mrtaurho Aug 27 '20 at 3:42

That's quite an impressive method to show that the integral vanishes.

For the first part I'll show using a different approach that your integral vanishes. $$\mathcal J=\int_0^1 \ln\left(\frac{x+3}{(x+2)(x+1)}\right)\frac{\mathrm dx}{x+1}\overset{x+1=t}= \color{blue}{\int_1^2\ln\left(\frac{t+2}{t+1}\right)\frac{\mathrm dt}{t}}-\color{red}{\int_1^2 \frac{\ln t}{t}\mathrm dt}$$ Let's denote the blue integral as $$\mathcal J_1$$ then using the substitution $$\frac{2}{t}\to t$$ we get: $$\mathcal J_1=\int_1^2 \ln\left(\frac{t+2}{t+1}\right)\frac{\mathrm dt}{t}=\int_1^2 \ln\left(\frac{2(t+1)}{t+2}\right)\frac{\mathrm dt}{t}$$ Adding both integrals from above gives us: $$\require{cancel} 2\mathcal J_1=\cancel{\int_1^2 \ln\left(\frac{t+2}{t+1}\right)\frac{\mathrm dt}{t}}+\int_1^2 \frac{\ln 2}{t}\mathrm dt+\cancel{\int_1^2 \ln\left(\frac{t+1}{t+2}\right)\frac{\mathrm dt}{t}}=\ln^2 2$$ $$\Rightarrow \mathcal J_1=\color{blue}{\frac{\ln^2 2}{2}}\Rightarrow \mathcal J=\color{blue}{\frac{\ln^2 2}{2}}-\color{red}{\frac{\ln^2 2}{2}}=0$$

As for the second part, a small generalization outcomes by experimenting with the blue integral.

In particular, by the same approach we have: $$\int_1^a \ln\left(\frac{x+a}{x+1}\right)\frac{\mathrm dx}{x}=\int_1^a \frac{\ln x}{x}\mathrm dx$$ Which gives us a small generalization: $${\int_0^{a-1}\ln\left(\frac{x+a+1}{(x+1)(x+2)}\right)\frac{\mathrm dx}{x+1}=0}$$ Similarly, (with the substitution $$\frac{ab}{x}\to x$$) we get that: $$\int_a^b \ln\left(\frac{x+b}{x+a}\right)\frac{dx}{x}=\frac12 \ln^2 \left(\frac{b}{a}\right)=\int_a^b \ln\left(\frac{x}{a}\right)\frac{dx}{x}$$ And the following follows: $$\int_{a-1}^{b-1} \ln\left(\frac{a(x+b+1)}{(x+1)(x+a+1)}\right)\frac{dx}{x+1}=0$$ One might be interested in the following similar generalization too: $$\int_1^{t}\ln\left(\frac{x^4+sx^2+t^2}{x^3+sx^2+t^2x}\right)\frac{dx}{x}=0,\quad s\in R, t>1$$

• This is what I was looking for; I guess^^ (+1) anyway and I'm looking forward to see a generalization (if possible). I guess you know where I got this integral from :D – mrtaurho Aug 12 '19 at 15:17
• @mrtaurho I got there a small generalization for now. However since $\int_a^b \ln\left(\frac{x+b}{x+a}\right)\frac{dx}{x}=\frac12 \ln^2 \left(\frac{b}{a}\right)$ it might be possible to obtain a better one. (I'll try later to work with it). – Zacky Aug 12 '19 at 15:38
• It's hard to write out your name now :P But yes, this seems promising, I'm curious! As I noted above it was a rather strange by-product to discover this equality; and it was tedious to ran into it three times while hoping for something more helpful for solving the original task^^' – mrtaurho Aug 12 '19 at 15:44
• @mrtaurho just in case you missed it in winter I'll mention that $\mathcal I$ (before the self-similar sub was applied) originates from this generalization: math.stackexchange.com/a/3049039/515527. Aka: $$\int_1^{\sqrt{t}}\ln\left(\frac{x^4+sx^2+t}{x(x^2+sx+t)}\right)\frac{dx}{x}=0$$ – Zacky Aug 12 '19 at 16:06
• As I've upvoted both, the question you linked and your answer, I guess I've seen it at some point. But I'll take a look at it again :) – mrtaurho Aug 12 '19 at 16:23

I have used the substitution $$(x+1)(y+1)=2$$ before to good effect because $$\int_0^1f(x)\,\frac{\mathrm{d}x}{x+1}=\int_0^1f\!\left(\tfrac{1-y}{1+y}\right)\frac{\mathrm{d}y}{y+1}\tag1$$ If $$f(x)=\log\left(\frac{x+3}{(x+2)(x+1)}\right)$$, then $$f\!\left(\frac{1-y}{1+y}\right)=\log\left(\frac{(y+2)(y+1)}{y+3}\right)$$. Therefore $$\int_0^1\log\left(\frac{x+3}{(x+2)(x+1)}\right)\frac{\mathrm{d}x}{x+1}=\int_0^1\log\left(\frac{(y+2)(y+1)}{y+3}\right)\frac{\mathrm{d}y}{y+1}\tag2$$ and since the two sides of $$(2)$$ are negatives, they are both $$0$$.
We can generalize $$(1)$$ by letting $$(x+a)(y+a)=a(1+a)$$, then we get $$\int_0^1f(x)\frac{\mathrm{d}x}{x+a}=\int_0^1f\!\left(\tfrac{a(1-y)}{a+y}\right)\frac{\mathrm{d}y}{y+a}\tag3$$