Finding $ \int^1_0 \frac{\ln(1+x)}{x}dx$

There is supposed to be a clean solution to the integral below, maybe involving some symmetry

$$ \int^1_0 \frac{\ln(1+x)}{x}dx$$

I have tried integration by parts as followed: $\ln(x+1)=u$ ,$\frac{1}{x+1} dx = du$ and also $\frac{1}{x}dx=dv$, $\ln(x)=v$. Then, the integral becomes

$$\int^1_0 \frac{\ln(1+x)}{x}dx=- \int^1_0 \frac{\ln x}{1+x}dx$$

which does not make this easier.

I have also tried using the identity $\int_a^bf(x)~dx=\int_a^bf(a+b-x)~dx$. So let $I = \int^1_0 \frac{\ln(1+x)}{x}dx$ and also $I = \int^1_0 \frac{\ln(2-x)}{1-x} $. Then $$2I= \int^1_0 \ln(1+x)+\ln(2-x)$$ which is not any easier, either.

Any ideas? :)

Solution 1:

Here is an elementary integration of $I=\int^1_0 \frac{\ln(1+x)}{x}dx $

\begin{align} I=-\frac12 \int^1_0 \underset{x\to x^3}{\frac{\ln(1+x)}{x}dx }+\frac32\int^1_0 \frac{\ln(1+x)}{x}dx =-\frac32\int^1_0 \frac{\ln(1-x+x^2)}{x}dx \end{align} Let $J(a)=\int^1_0 \frac{\ln(1-2x\sin a+x^2)}{x}dx$ $$J’(a)=-\int^1_0 \frac{2\cos a\ }{(x-\sin a)^2 +\cos^2a}dx =-\left(\frac\pi2+a\right)$$

Then, with $J(0)=\int^1_0 {\frac{\ln(1+x^2)}{x}dx } \overset{x^2\to x} =\frac12I $

$$I= -\frac32J(\frac\pi6)= -\frac32\left( J(0)+\int_0^{\frac\pi6}J’(a)da\right)=-\frac34I +\frac32\int_0^{\frac\pi6}\left(\frac\pi2+a\right)da $$ which leads to

$$I= \frac{\pi^2}{12}$$

Solution 2:

Hint. One may recall that $$ \ln (1+x)=\sum_{n=1}^\infty(-1)^{n-1}\frac{x^n}{n},\quad |x|<1, $$ one may then divide by $x$ and one is allowed to integrate termwise obtaining $$ \int_0^1\frac{\ln (1+x)}x\:dx=\sum_{n=1}^\infty(-1)^{n-1}\int_0^1\frac{x^{n-1}}{n}=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^2}. $$ Can you take it from here?

Solution 3:

By integration by parts: $$ \int_{0}^{1}\frac{\log(1+x)\,dx}{x} = -\int_{0}^{1}\frac{\log x}{1+x}\,dx=2\int_{0}^{1}\frac{-\log x}{1-x^2}\,dx-\int_{0}^{1}\frac{-\log x}{1-x} $$ but since $\int_{0}^{1}(-\log x)x^k\,dx = \frac{1}{(k+1)^2}$, by expanding $\frac{1}{1-x^2}$ and $\frac{1}{1-x}$ as geometric series we get: $$ \int_{0}^{1}\frac{\log(1+x)\,dx}{x} = 2\sum_{k\text{ odd}}\frac{1}{k^2}-\sum_{k\geq 1}\frac{1}{k^2}=\sum_{k\geq 1}\frac{1}{k^2}-2\sum_{k\text{ even}}\frac{1}{k^2}=\frac{1}{2}\sum_{k\geq 1}\frac{1}{k^2}=\color{blue}{\frac{\pi^2}{12}}. $$

Solution 4:

$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ \begin{align} \int_{0}^{1}{\ln\pars{1 + x} \over x}\,\dd x & \,\,\,\stackrel{x\ \mapsto\ -x}{=}\,\,\, \int_{0}^{-1}{\ln\pars{1 - x} \over x}\,\dd x = -\int_{0}^{-1}\mrm{Li}_{2}'\pars{x}\,\dd x = -\mrm{Li}_{2}\pars{-1} + \mrm{Li}_{2}\pars{0} \\[5mm] & = \bbx{\ds{\pi^{2} \over 12}} \\ & \end{align}