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1500 questions
81
votes
1 answer
The closed form of $\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$
What tools or ways would you propose for getting the closed form of this integral?
$$\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$$
EDIT: It took a while since I made this post. I'll give a little bounty for the solver of the problem,…
user 1591719
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- 255
81
votes
3 answers
Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$
Using $\text{n}^{\text{th}}$ root of unity
$$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$
Prove that
$$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
Ali_ilA
- 813
81
votes
4 answers
Proving that the tensor product is right exact
Let
$A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0$ a exact sequence of left $R$-modules and $M$ a left $R$-module ($R$
any ring).
I am trying to prove that the induced sequence $$A\otimes_R…
Klaus
- 4,105
81
votes
7 answers
Hanging a picture on the wall using two nails in such a way that removing any nail makes the picture fall down
A friend of mine told me that it's possible to hang a picture on the wall from a string using two nails in such a way that removing either of the two nails will make both the string and picture fall down. My friend also told me that I need to be…
Anonymous - a group
- 1,463
81
votes
1 answer
Theorem that von Neumann proved in five minutes.
In "How To Solve It", George Pólya writes:
"There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain
theorem, and I said it is not proved and it may be difficult. Von
Neumann…
Salech Alhasov
- 6,780
- 2
- 29
- 47
81
votes
14 answers
Are all mathematicians human calculators?
I asked my dad why he did not major in math he said "because he is not good at math".
I think I like math, and I think I'm ok at it, but I'm not gifted or anything like that, I just like math. I think I'd like to major in math, but I see all these…
Kat
- 1,371
81
votes
10 answers
Finding out the area of a triangle if the coordinates of the three vertices are given
What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane?
One approach is to find the length of each side from the coordinates given and then apply Heron's formula. Is this the…
TSP1993
- 969
- 1
- 9
- 11
81
votes
8 answers
How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x - \frac{1}{x}\right)dx?$
If $f(x)$ is a continuous function on $(-\infty, +\infty)$ and $\int_{-\infty}^{+\infty} f(x) \, dx$ exists. How can I prove that
$$\int_{-\infty}^{+\infty} f(x) \, dx = \int_{-\infty}^{+\infty} f\left( x - \frac{1}{x} \right) \, dx\text{ ?}$$
Hung Nguyen
- 1,841
81
votes
1 answer
Compute $ \lim\limits_{n \to \infty }\sin \sin \dots\sin n$
I need your help with evaluating this limit:
$$ \lim_{n \to \infty }\underbrace{\sin \sin \dots\sin}_{\text{$n$ compositions}}\,n,$$
i.e. we apply the $\sin$ function $n$ times.
Thank you.
user6163
81
votes
1 answer
Thurston's 37th way of thinking about the derivative
In Thurston's superb essay On proof and progress in mathematics, he makes this observation:
Of course there is always another subtlety to be gleaned, but I would like to at least think that I have absorbed the main intuition behind each element…
Zev Chonoles
- 129,973
81
votes
1 answer
Does every Cauchy sequence converge to *something*, just possibly in a different space?
Question. If I attempt to prove that space $X$ is complete by pursuing the strategy, “Assume $x_n \rightarrow x$; the space $X$ is complete if $x \in X$,” then why is that wrong?
Context. I know the definition of Cauchy sequences and convergent…
1Teaches2Learn
- 2,439
81
votes
3 answers
Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)
I am trying to understand the differences between
$$
\begin{array}{|l|l|l|}
\textbf{vector space} & \textbf{general} & \textbf{+ completeness}\\\hline
\text{metric}& \text{metric space} & \text{complete space}\\
\text{norm} & \text{normed} &…
vonjd
- 8,810
81
votes
13 answers
What is an example of a sequence which "thins out" and is finite?
When I talk about my research with non-mathematicians who are, however, interested in what I do, I always start by asking them basic questions about the primes. Usually, they start getting reeled in if I ask them if there's infinitely many or not,…
tomos
- 1,650
81
votes
7 answers
What is lost when we move from reals to complex numbers?
As I know when you move to "bigger" number systems (such as from complex to quaternions) you lose some properties (e.g. moving from complex to quaternions requires loss of commutativity), but does it hold when you move for example from naturals to…
Юрій Ярош
- 2,086
81
votes
4 answers
Why did my friend lose all his money?
Not sure if this is a question for math.se or stats.se, but here we go:
Our MUD (Multi-User-Dungeon, a sort of textbased world of warcraft) has a casino where players can play a simple roulette.
My friend has devised this algorithm, which he himself…
Konerak
- 1,043