Evaluating $\int_{0}^{1}\frac{1-x}{1+x}\frac{\mathrm dx}{\ln x}$

Some time ago I came across to the following integral:

$$I=\int_{0}^{1}\frac{1-x}{1+x}\frac{\mathrm dx}{\ln x}$$ What are the hints on how to compute this integral?

• What happens if you substitute $u=\ln x$? (On second thought, it doesn't look too useful.) – Harald Hanche-Olsen Jan 16 '13 at 15:30
• @HaraldHanche-Olsen: Isn't it an improper integral? – Mikasa Jan 16 '13 at 15:39
• Fwiw Wolfram alpha couldn't find an indefinite integral, although it can find the definite integral, presumably by approximation. – JSchlather Jan 16 '13 at 15:40
• @JacobSchlather: But it converges. – Mikasa Jan 16 '13 at 15:40
• Maple says the value is $-\ln(\pi/2)$. – Harald Hanche-Olsen Jan 16 '13 at 15:42

Make the substitution $x=e^{-y}$ and do a little algebra to get the value of the integral to be

$$\int_0^{\infty} \frac{dy}{y} \frac{e^{-y} - e^{-2 y}}{1+e^{-y}}$$

Now Taylor expand the denominator and get

$$\int_0^{\infty} \frac{dy}{y} (e^{-y} - e^{-2 y}) \sum_{k=0}^{\infty} (-1)^k e^{-k y}$$

If we can reverse the order of sum and integral, we get

$$\sum_{k=0}^{\infty} (-1)^k \int_0^{\infty} \frac{dy}{y} (e^{-(k+1) y} - e^{-(k+2) y})$$

The integral inside may be evaluated exactly, and the result is the sum

$$\sum_{k=0}^{\infty} (-1)^k \log{\frac{k+1}{k+2}}$$

$$= \lim_{n \rightarrow \infty} \; \log{\frac{\frac{1}{2} \frac{3}{4} \ldots \frac{2 n-1}{2 n}}{\frac{2}{3} \frac{5}{6} \ldots \frac{2 n+1}{2 n+2}}}$$

$$= \log{\left ( \frac{2}{\pi} \right )}$$

• Seems reasonable. Thank you! – Martin Gales Jan 16 '13 at 16:27
• Thank you for asking it - this was fun! – Ron Gordon Jan 16 '13 at 16:40
• Yeah, math is about having fun! :-) (+1) – user 1591719 Jan 16 '13 at 16:54
• To reverse the order of sum and integra, I understand that you appeal to the absolute convergence of sum and integral, that is, to the convergence of the positive series $\sum\limits_{k\geqslant0}\int\limits_0^{+\infty}(e^{-(k+1)y}-e^{-(k+2)y})dy/y$. But this series diverges. Maybe I did not understand the result you are applying, could you explain? – Did Jan 21 '13 at 9:17
• Maybe we are speaking across each other. Look, do you agree that the sum in Line 2 absolutely converges (in this case, to $(1-e^{-y})^{-1})$? And that the integral absolutely converges (in this case, to $\log{\frac{k+1}{k+2}}$). Because each integral is also continuous, then we get a convergent result in exchanging the order of the sum and integral in Line 2. As I said, this does not apply to Line 3 because, as you point out, that the sum does not converge absolutely. But I am not seeking to reverse the order of integration here, just in Line 2. – Ron Gordon Jan 21 '13 at 14:38

I tried with "Differentiation under integration sign":

$$J(\alpha)=\int_0^1\frac{1-x^\alpha}{1+x}\frac{dx}{\ln x}\quad \alpha>0$$

Then, as usual, $$\frac{dJ}{d\alpha}=-\int_0^1\frac{x^\alpha}{1+x}dx=-f(\alpha),\text{(say)}$$

Then, integrating $$J(\alpha)=-\int f(\alpha)d\alpha+c$$ I used Mathematica to evaluate $f(\alpha)$ (it gives difference of two Harmonic numbers) and its integral. The integral is just $\ln\frac{\Gamma(\frac{2m+1}{2})}{\Gamma(\frac{m+1}{2})}$. Note that $J(0)=0$ and putting $\alpha=1$, the result follows.

• Very interesting approach! Thanks! – Martin Gales Jan 17 '13 at 16:05
• Where does $m$ come from in $\ln\dfrac{\Gamma(\frac{2m+1}{2})}{\Gamma(\frac{m+1}{2})}$? – Alex Schiff Aug 1 '14 at 14:33

Consider $$\mathcal{I}(\alpha)=\int_0^1\frac{1-x^\alpha}{(1+x)\ln x}\ dx.\tag1$$ Differentiating $(1)$ with respect $\alpha$ yields \begin{align} \frac{d\mathcal{I}}{d\alpha}&=\int_0^1\frac{\partial}{\partial\alpha}\left[\frac{1-x^\alpha}{(1+x)\ln x}\right]\ dx\\ &=-\int_0^1\frac{x^\alpha}{1+x}\ dx\\ &=-\int_0^1\sum_{k=0}^\infty(-1)^kx^{\alpha+k}\ dx\\ &=-\sum_{k=0}^\infty(-1)^k\int_0^1x^{\alpha+k}\ dx\\ &=-\sum_{k=0}^\infty\frac{(-1)^k}{\alpha+k+1}.\tag2 \end{align} Now consider polygamma function $$\psi_n(z)=\frac{d^{n+1}}{dz^{n+1}}\ln\Gamma(z)\tag3$$ and $$\psi_n(z)=(-1)^{n+1}n!\sum_{k=0}^\infty\frac{1}{(z+k)^{n+1}}.\tag4$$ Hence by using $(4)$ we obtain $$\sum_{k=0}^\infty\frac{(-1)^{k}}{(z+k)^{n+1}}=\frac1{(-2)^{n+1}n!}\left[\psi_n\left(\frac{z}{2}\right)-\psi_n\left(\frac{z+1}{2}\right)\right].\tag5$$ Using $(3)$ and $(5)$ then $(2)$ becomes \begin{align} \frac{d\mathcal{I}}{d\alpha}&=\frac12\left[\psi_0\left(\frac{\alpha+1}{2}\right)-\psi_0\left(\frac{\alpha+2}{2}\right)\right]\\ \mathcal{I}(\alpha)&=\frac12\int\left[\psi_0\left(\frac{\alpha+1}{2}\right)-\psi_0\left(\frac{\alpha+2}{2}\right)\right]\ d\alpha\\ &=\ln\Gamma\left(\frac{\alpha+1}{2}\right)-\ln\Gamma\left(\frac{\alpha+2}{2}\right)+C\\ &=\ln\Gamma\left(\frac{\alpha+1}{2}\right)-\ln\Gamma\left(\frac{\alpha+2}{2}\right)-\frac12\ln\pi,\tag6 \end{align} where $\mathcal{I}(0)=0$ and $C=-\ln\Gamma\left(\frac{1}{2}\right)$.

Thus \begin{align} \int_0^1\frac{1-x}{(1+x)\ln x}\ dx&=\mathcal{I}(1)\\ &=\ln\Gamma\left(1\right)-\ln\Gamma\left(\frac{3}{2}\right)-\frac12\ln\pi\\ &=\color{blue}{\ln\left(\frac{2}{\pi}\right)}. \end{align}


You may like this method. Note $$\int_0^1x^ada=-\frac{1-x}{\ln x}, \text{ for }x>0, H_a=\sum_{k=1}^\infty\frac{a}{k(a+k)},$$ and hence \begin{eqnarray} \int_0^1\frac{1-x}{1+x}\frac{dx}{\ln x}&=&-\int_0^1\int_0^1\frac{x^a}{1+x}dadx\\ &=&-\int_0^1\int_0^1\frac{x^a}{1+x}dxda\\ &=&-\frac{1}{2}\int_0^1(H_{\frac{a}{2}}-H_{\frac{a-1}{2}})da\\ &=&-\frac{1}{2}\int_0^1\sum_{k=1}^\infty\left(\frac{a}{k(a+2k)}-\frac{a-1}{k(a+2k-1)}\right)da\\ &=&-\frac{1}{2}\sum_{k=1}^\infty\left(4\ln k-2\ln(k^2-\frac{1}{4})\right)\\ &=&-\frac{1}{2}\ln\prod_{k=1}^\infty\frac{k^4}{(k^2-\frac{1}{4})^2}\\ &=&-\prod_{k=1}^\infty\frac{k^2}{k^2-\frac{1}{4}}\\ &=&-\ln\left(\frac{\pi}{2}\right). \end{eqnarray} In the last step, we used the Wallis Formula from here.