0
This question (I'd never close or put on hold) also has the value of an answer to the first integral here How to evaluate $\int_0^1\frac{\ln(x)\ln(1+x^2)}{1-x}dx$ and $\int_0^1\frac{\ln(x)\ln(1+x^2)}{1+x}dx$
$$\int_0^1\frac{\ln(x)\ln(1+x^2)}{1-x}dx$$
$$=\int_0^1\frac{(1+x)(1+x^2) \log(x)\log(1-x...
Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this result?
2
Let $\sqrt{1+x}-\sqrt{1-x}\mapsto2\sin{x}$. We get, for the first integral,
\begin{align}
\color{#6F00FF}{\int^1_0\ln^2\left(\sqrt{1+x}-\sqrt{1-x}\right)\ {\rm d}x}
=&\int^\frac{\pi}{4}_02\cos{2x}\ln^2(2\sin{x})\ {\rm d}x\\
=&\sin{2x}\ln^2(2\sin{x})\Bigg{|}^\frac{\pi}{4}_0-\color{#FF4F00}{\int^\f...
In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept of mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof: a mistake in a proof leads to an invalid proof just in the same way, but in the best-known examples of mathematical fallacies, there is some concealment in the presentation of the proof. For example, the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as typically presented...
In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept of mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof: a mistake in a proof leads to an invalid proof just in the same way, but in the best-known examples of mathematical fallacies, there is some concealment in the presentation of the proof. For example, the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as typically presented...
