<s>[INST] How to evaluate $\int_{0}^{1} \frac{\ln x}{x+1} dx$ I want to evaluate:
$$\int_{0}^{1} \frac{\ln x}{x+1} dx$$
If I was asked I would to evaluate:
$$\int_{0}^{1} \frac{\ln x}{x-1} dx$$
That would be easy because if I use the Taylor series for $\ln x$ centered at $1$ then things will cancel out and leave me with a easy integral.
So how about this integral, I'm guessed to use  the Taylor series centered around $-1$. But even with that thought in mind,  it does not take me anywhere because $\ln (-1)$ is not defined. Can someone help.
 [/INST] Consider:
$$I=\int_{0}^{1}\frac{\ln(x)}{1+x}dx= \sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}$$
You can show this result by noticing since $0<x<1$ that $$\frac{1}{1+x}= \sum_{n=0}^{\infty}(-x)^n$$ and performing term by term integration. You will need integration by parts to do that.
Now you can also show that:
$$-I=\int_{0}^{1}\int_{0}^{1}\frac{1}{1+xy}dydx$$ by expanding the integrand into a geometric series and doing term by term integration with respect to $y$ and the with respect to $x$.
Now, in the  double integral, let $x=u+v,y=-u+v$
Then $\left|\frac{\partial(x,y)}{\partial(u,v)}\right|=\begin{vmatrix} 1 && 1\\ -1 && 1 \\ \end{vmatrix}=2$
The change of variables causes the double integral to become:
$$\iint_{T} \frac{2}{1+v^2-u^2}dvdu$$ where $T$ is the square with vertices $(0,0),(\frac{-1}{2},\frac{1}{2}),(0,1),(\frac{1}{2},\frac{1}{2})$
Now, we need to compute two double integrals (over two different subregions of the square) and sum them to get the final answer.
First let $\frac{-1}{2}\leq u \leq 0 , -u \leq v \leq 1+u$. 
$$\int_{\frac{-1}{2}}^{0}\int_{-u}^{1+u} \frac{2}{1+v^2-u^2}dvdu=\int_{\frac{-1}{2}}^{0}\frac{2\tan^{-1}\left(\frac{1+u}{\sqrt{1-u^2}}\right)+2\tan^{-1}\left(\frac{u}{\sqrt{1-u^2}}\right)}{\sqrt{1-u^2}}du$$
For the first term on the right hand side, let : $z=2\tan^{-1}\left(\frac{1+u}{\sqrt{1-u^2}}\right),dz=\frac{1}{\sqrt{1-u^2}}du$. For the second term, if you draw a right triangle, notice $2\tan^{-1}\left(\frac{u}{\sqrt{1-u^2}}\right)=2\sin^{-1}\left(u\right)$ so let $z=\sin^{-1}\left(u\right),dz=\frac{1}{\sqrt{1-u^2}}du$ for the second term.When evaluating these two terms with these substitutions, the sum of these terms should be $\frac{\pi^2}{24}$.
On the other hand, let $0\leq u \leq \frac{1}{2} , u \leq v \leq 1-u$. 
$$\int_{0}^{\frac{1}{2}}\int_{u}^{1-u} \frac{2}{1+v^2-u^2}dvdu=\int_{0}^{\frac{1}{2}}\frac{2\tan^{-1}\left(\frac{1-u}{\sqrt{1-u^2}}\right)-2\tan^{-1}\left(\frac{u}{\sqrt{1-u^2}}\right)}{\sqrt{1-u^2}}du$$
For the first term on the right hand side, let $z=2\tan^{-1}\left(\frac{1-u}{\sqrt{1-u^2}}\right),dz=\frac{-1}{\sqrt{1-u^2}}$ and for the second term, use $2\tan^{-1}\left(\frac{u}{\sqrt{1-u^2}}\right)=2\sin^{-1}\left(u\right)$ and let $z=\sin^{-1}\left(u\right),dz=\frac{1}{\sqrt{1-u^2}}du$ for this.When evaluating these two terms with these substitutions, the sum of these terms should be $\frac{\pi^2}{24}$.
So:
$$\iint_{T} \frac{2}{1+v^2-u^2}dvdu=\frac{2\pi^2}{24}=\frac{\pi^2}{12}$$
thus, $$I=\int_{0}^{1}\frac{\ln(x)}{1+x}dx=\frac{-\pi^2}{12}$$
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