Most active questions
895 questions from the last 7 days
68
votes
2
answers
2k
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coincidental (?) patterns in logs of repeating decimals, e.g. $\ln(2/3)$ vs. $\ln(0.6666666)$
I was playing with logarithms under an arbitrary-precision calculator, and got some odd results. I happened to have the precision set to 20 places, and these are the initial results I got:
...
- 699
13
votes
5
answers
331
views
How to compute $\int_0^\infty\frac{dx}{(x^2+1)\sqrt{x^2-x+1}}$?
This integral appeared in my recent integral calculus test. I tried to make a few attempts, but none of them seemed to simplify the integral.
Attempt $1$: Substituting $x\to\tan x$,
$$I=\int_0^{\pi/2}\...
27
votes
1
answer
2k
views
Is there a "true" value of BB(745)?
I was reading this reddit thread and I got confused by one part. I always thought that there is always a "true" value of BB(n), even though it might not be provable or findable.
There is a ...
- 6,811
5
votes
3
answers
1k
views
What is the probability that the graph remains connected?
Consider the complete graph $K_4$ with four vertices; all vertices are connected by an edge to all other vertices. Suppose now that we flip an unbiased coin for each edge. If heads comes up, we leave ...
- 357
8
votes
8
answers
285
views
How to evaluate $\lim\limits_{x \to 0^{+}} \frac{\sqrt{\sin x} - \sin \sqrt{x}}{x}$?
I am trying to evaluate the following limit:
$$\lim\limits_{x \to 0^{+}} \frac{\sqrt{\sin x} - \sin \sqrt{x}}{x}$$
Kindly guide as I have made no significant progress in this.
I tried the L'Hospital's ...
- 517
9
votes
3
answers
444
views
If a stochastic matrix has unit permanent, is it a permutation matrix?
In this question, a stochastic square matrix is a real square matrix where all the rows sum up to $1$ and all the entries are between $0$ and $1$. Permutation matrices are examples of stochastic ...
- 327
5
votes
3
answers
738
views
Least naturals with square average and cube product: $m+n = 2j^2, mn = k^3$
Say that two distinct natural numbers $1\le n < m$ of the same parity are harmonious if their mean $(n + m)/2$ is a perfect square and their product $nm$ is a perfect cube. For instance $(n,m ) = (...
- 502
6
votes
4
answers
199
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Is there any alternative to evaluate $\int_0^1\left(\sin ^{-1} x\right)^2 \ln x \,\mathrm dx$?
When I encountered the integral
$$
I=\int_0^1\left(\sin ^{-1} x\right)^2 \ln x \,\mathrm dx,
$$
I just wondered whether it can be solved by some “elementary” methods. I tried the substitution $y=\sin^{...
- 29.5k
8
votes
2
answers
714
views
Are there examples of functions that can be quadratically aproximated at certain points but not diffrentiable twice at those points?
I am going over calculus 3 (in a different language which may explain wrong terms used which I would appreciate if corrected).
I got to the point where we are trying to understand the intuition of ...
- 327
3
votes
4
answers
210
views
What allows us to go from $3^{3x} = 3^9$ to $3x = 9$?
With $3^{3x} = 3^9$, I can simply jump to $3x = 9$, or $x = 3$.
It's staring me in the face obvious, yes, but what is the "official" algebra allowing us to "throw away" the base $3$...
- 1,162
7
votes
3
answers
211
views
Is it hard to evaluate the integral $\int_0^{\frac{\pi}{4}} \tan ^{-1} \sqrt{\frac{1-\tan ^2 x}{2}} d x$ without Feynman’s trick?
Once I was attracted by the decent value of the integral$$
I=\int_0^{\frac{\pi}{4}} \tan ^{-1} \sqrt{\frac{1-\tan ^2 x}{2}} d x=\frac{\pi^2}{24},
$$
I attempted to get rid of the surd in the integral ...
- 29.5k
4
votes
2
answers
648
views
Would we consider a point circle to be a "real circle"?
Well, I came across this question in my exam:
Let the variable real circle $S:x^2+y^2-ax-by+c=0$ intersects the pair of straight lines $xy-4x-3y+12=0$ orthogonally and the circle lies in the first ...
- 57
3
votes
4
answers
266
views
Computing a definite integral: $\frac{1}{\pi^2} \int_0^\infty \frac{(\ln x)^2}{\sqrt{x}(1 - x)^2} dx$ [duplicate]
This is a question my friend and I found on a Telegram channel, and I'm having trouble solving it.
The value of the definite integral
$$\frac{1}{\pi^2} \int_0^\infty \frac{(\ln x)^2}{\sqrt{x}(1 - x)^...
- 318
12
votes
1
answer
633
views
+100
Calculating $\int_0^1 \frac{\arctan^3(x) \log(x)} x\,\mathrm dx$
I’m looking for various ways of either calculating or reducing the integral to simple, related harmonic series,
$$\int_0^1 \frac {\arctan^3(x) \log(x)} x \,\mathrm dx $$
$$= \frac {121} {15360} \pi^5 +...
- 6,567
5
votes
2
answers
541
views
All subspaces Hausdorff then Hausdorff?
Let $X$ be a topological space with $|X|>2$ satisfying
(*) For any proper $Y\subsetneq X$, the subspace $Y$ is Hausdorff.
Is it necessary that $X$ is also Hausdorff?
I can only do this by ...
- 170