# Prove: $\int_0^1 \frac{\ln x }{x-1} d x=\sum_1^\infty \frac{1}{n^2}$

I'd like your help with proving that $$\int_0^1 \frac{\ln x }{x-1}d x=\sum_{n=1}^\infty \frac{1}{n^2}.$$ I tried to use Fourier series, or to use a power series and integrate it twice but it didn't work out for me.

Any suggestions?

Thanks!

• Hint: take the derivative of $\ln{x}\ln(1-x)$, and remember that $x\ln{x}\rightarrow{0}$ as $x\rightarrow{0}$. Feb 11, 2012 at 20:31
• Fiddling with this led me to think about $\lim\limits_{x\to0+}((\log x)(\log(1-x)))$. Maybe if I ever have a class of bright students taking calculus, I'll assign this. (...and I see "bgins" also thought of that.) Feb 11, 2012 at 20:38
• @bgins: Can you please extend your comment? I'm not sure I understand what to do. Feb 11, 2012 at 20:46
• @Jozef: $[\ln(x)\ln(1-x)]'=\frac{\ln(1-x)}{x}+\frac{\ln{x}}{x-1}$. Integrating, the LHS is zero because of the limit mentioned. Then, we have the same situation as americo-tavares and peter below, except their routes are better because more they are direct (no pulling rabbits out of hats)! Feb 11, 2012 at 22:17
• @AméricoTavares I'm not sure (I shouldn't have said "needed")... The fact in my previous comment was the only theorem I could find that justifies the method. I'd be interested to know the answer to your question too. Feb 11, 2012 at 23:52

Hint: use the substitution $u=1-x$ to obtain $$I:=\int_{0}^{1}\frac{\ln x}{x-1}dx=-\int_{0}^{1}\frac{\ln \left( 1-u\right) }{u}\,du$$

and the following Maclaurin series $$\ln \left( 1-u\right) =-u-\frac{1}{2}u^{2}-\frac{1}{3}u^{3}-\ldots -\frac{ u^{n+1}}{n+1}-\ldots\qquad(\left\vert u\right\vert <1)$$

• In case anybody reads this answer, I think it is necessary to say something like for all $u\in(0,1)$ and for all $n\in\Bbb N$, $$\left|1+\frac12u+\frac13u^2+\cdots\frac1nu^{n-1}\right|\leq-\frac{\ln(1-u)}u=g(u)$$ so that, since $g$ is integrable on $(0,1)$, one can apply dominated convergence, and swap the sum with the integral. Jan 19, 2015 at 19:15
• @Olivier Bégassat Thanks for your comment. I've just assumed, without justification, that the given integral could be evaluated by expanding the integrand in a series and integrating term by term. Jan 19, 2015 at 19:48

$$\int_0^1 \frac{\log x}{x-1}dx =\lambda$$

Making $x = 1-u$ produces (keep the $x$)

$$-\int_0^1 \frac{\log (1-x)}{x}dx=\lambda$$

$$\frac{\log (1-x)}{x}=-\sum_{n=1}^{\infty} \frac{x^{n-1}}{n}$$

$$-\int_0^1 \frac{\log (1-x)}{x}dx =\left.\sum_{n=1}^{\infty} \frac{x^{n}}{n^2} \right|_0^1 =\sum_{n=1}^{\infty} \frac{1}{n^2}$$

• This is exactly Américo's answer (in fact all alnswers are the same :-) ) Jan 15, 2013 at 4:26

Write $\ln x = \ln(1 + (x-1))$ and use the log series

Related problem: I, II. Using the change of variables $$u=-\ln(x)$$ and the identity

$$\int_{0}^{\infty}\frac{u^{s-1}}{e^u -1}=\zeta{(s)}\Gamma{(s)}$$

we reach to the deisred result

$$\int_0^1 \frac{\ln x }{x-1}= \int_{0}^{\infty}\frac{u}{e^u -1}=\zeta{(2)}\Gamma{(2)} =\sum_{n=1}^\infty \frac{1}{n^2}.$$

$$\int_{0}^{\infty}\frac{u^{s-1}}{e^u - 1}=\int_{0}^{\infty}\frac{u^{s-1}}{e^u}(1-e^{-u})^{-1}= \sum_{n=0}^{\infty} \int_{0}^{\infty}{u^{s-1}e^{-(n+1)u}}$$

$$= \sum_{n=0}^{\infty}\frac{1}{(n+1)^s} \int_{0}^{\infty}{y^{s-1}e^{-y}}= \sum_{n=1}^{\infty}\frac{1}{n^s} \Gamma(s)= \zeta(s) \Gamma(s).$$

• Where does the result for this last integral come from?
– Alex
Feb 21, 2013 at 21:52
• @Alex: See the added. Feb 24, 2013 at 16:06

Hint: Use a geometric sum and a partial integration $$\int_0^1x^n\log x \,dx=\frac{x^{n+1}}{n+1}\log x \bigg|_0^1-\int_0^1\frac{x^{n}}{n+1}$$

Edit: The first step is $$\frac{\log x}{x-1}=-\frac{\log x}{1-x}=-\log x\sum_{k=0}x^k$$

Using the expansion series $$\displaystyle\frac 1{1-x}=\displaystyle\sum_{n=0}^\infty x^n,\, |x|<1$$, we get $$\begin{eqnarray*} \int_0^1\frac{x^a}{x-1}\, dx&=& -\int_0^1 x^a\left(\sum_{n=0}^\infty x^{n}\, dx\right)\\ &=& -\sum_{n=0}^\infty\left(\int_0^1 x^{n+a}\, dx\right)\\ &=& -\sum_{n=0}^\infty \frac 1{(n+a+1)}\\ &=& -\sum_{n=1}^\infty \frac 1{(n+a)} \end{eqnarray*}$$ Differentiate wrt $$a$$, we obtain $$\begin{eqnarray*} \int_0^1\frac{x^a\ln(x)}{x-1}\, dx &=& \sum_{n=1}^\infty \frac 1{(n+a)^2} \end{eqnarray*}$$ Fix $$a=0$$ in both sides, we have $$\begin{eqnarray*} \int_0^1\frac{\ln(x)}{x-1}\, dx &=& \sum_{n=1}^\infty \frac 1{n^2}\\ &=&\frac{\pi^2}6. \end{eqnarray*}$$