User TheSimpliFire

1 vote

Evaluating $\int_{-\infty}^{+\infty}\frac{\sin{(\cosh{x})}\cosh{x}}{1+\cosh^{2}{x}}dx$

answered May 30 at 23:21

6 votes

Accepted

What is a good category for probability theory?

answered May 28, 2021 at 20:11

1 vote

What is the shape of the perfect coffee cup for heat retention assuming coffee is being drunk at a constant rate?

answered May 20 at 22:28

2 votes

Representation of integers by $A = B^{\frac{5}{4}} + C^{\frac{5}{4}}$

answered May 17 at 23:48

4 votes

Accepted

Asymptotic behavior of $\int_{0}^r \frac{\sin(x)^2}{x^2} x^{\alpha}dx$.

answered Apr 19 at 11:39

0 votes

Showing an integral is equal to arcsine

answered Mar 27 at 1:15

7 votes

Accepted

Matrix performing local differintegral analysis being its own inverse. Coincidence?

answered Jan 27 at 1:39

14 votes

Accepted

What is the value of $\lim\limits_{n\to\infty}2^{n}x_{n}$ if $x_1\in(0,1)$ and $x_{n+1}=\frac{\sqrt{1+x_n}-\sqrt{1-x_n}}2$ for every $n\ge1$?

answered Jan 28 at 18:42

19 votes

Accepted

Does there exist a real function with domain $\Bbb{R}$ such that $f'(x)>0$ and $f''(x)+(f'(x))^2<0$ for all $x$?

answered Jul 3, 2021 at 10:10

6 votes

Accepted

How I cut my orange - spherical volume integral

answered Dec 26, 2022 at 19:48

5 votes

$\int_0^\infty \log\left(1 - \frac{x}{\sinh x}\right)\,dx$ and generalisations

answered Nov 2, 2022 at 0:10

5 votes

Accepted

Show that $\int_0^\pi\int_0^\pi\frac{\sin(x) \sin(px) \cos(qy)}{\sin(x)^2 + \sin(y)^2}\,dx\,dy$ tends to $0$ as $p\to\infty$ or $q\to\infty$

answered Oct 24, 2020 at 14:36

0 votes

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Calculating $\operatorname{cov}(\lambda\sinh(\frac{z-\gamma}{\delta})+\xi, \sigma z+\mu)$

answered Sep 26, 2022 at 16:45

5 votes

Analogs of $\sum_{n\ge1}\frac{n^{13}}{e^{2\pi n}-1}=\frac{1}{24}$

answered Aug 2, 2022 at 20:30

6 votes

Accepted

Is it true that $(2^\ell-1)\sum_{k=1}^\ell\binom\ell{k}(\frac1n)^{2k}(1-\frac1n)^{2\ell-2k}+2(1-\frac1n)^\ell-1-(1-\frac1n)^{2\ell}\geq0$?

answered Jul 31, 2022 at 21:18

1 vote

Prove $\log(x + 1) \leq x - \frac{x^{2}}{a}$ for fixed $a > 0$

answered Jul 19, 2022 at 21:39

1 vote

Accepted

Problem with Veblen's proof for the transcendence of $\pi$

answered Jun 17, 2022 at 9:31

1 vote

Accepted

The integral $\int_{0}^{\pi/2} \tan^p x~dx$

answered May 30, 2022 at 19:31

8 votes

Can $\operatorname{Re}(a+bi)^{n}$ be overlapped with $a,b\in\mathbb{Z}$ fixed?

answered Feb 1, 2020 at 17:25

18 votes

Accepted

Is this a new characteristic function for the primes?

answered Oct 12, 2019 at 13:31

2 votes

Accepted

Find the smallest value of $\alpha\in \mathbb{R}$ such that for all $x>0$ you have $\left(1+\frac{1}{x}\right)^{x+\alpha}>e$

answered May 4, 2022 at 7:30

10 votes

Accepted

Rotation of Hyperbola with any angle

answered Sep 18, 2018 at 19:29

1 vote

An integration to first order

answered Apr 24, 2022 at 16:22

6 votes

Prove that $\int_0^\infty \frac{1+2\cos x+x\sin x}{1+2x\sin x +x^2}dx=\frac{\pi}{1+\Omega}$ where $\Omega e^\Omega=1$

answered Apr 22, 2022 at 12:23

9 votes

Accepted

Finding $\sum_{n=1}^{\infty}\frac{(-1)^n (H_{2n}-H_{n})}{n2^n \binom{2n}{n}}$

answered Apr 9, 2022 at 11:00

7 votes

How to evaluate $\sum_{k=1}^{\infty}\frac{B\left(k, \frac{1}{2}\right)}{(2k+1)^2}$, where $B(x, y)$ is the Beta function?

answered Apr 3, 2022 at 13:20

1 vote

How to solve $\sqrt{x+2}\geq x$?

answered Apr 1, 2022 at 11:16

8 votes

Prove that $\int_0^\frac{\pi}{4}\frac{\cos (n-2)x}{\cos^nx}dx=\frac{1}{n-1}2^\frac{n-1}{2}\sin\frac{(n-1)\pi}{4}$

answered Mar 28, 2022 at 6:41

5 votes

Number of zeros of $f(x)= \frac{1}{2} E\left[ \tanh \left( \frac{x+Z}{2} \right) \right]-\tanh(x)+\frac{x}{2}$ where $Z$ is standard normal

answered Jan 8, 2022 at 22:03

3 votes

Accepted

Can this formula for $\zeta(3)$ be proven or simplified further?

answered Feb 11, 2022 at 10:55

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808 Answers

Evaluating $\int_{-\infty}^{+\infty}\frac{\sin{(\cosh{x})}\cosh{x}}{1+\cosh^{2}{x}}dx$

What is a good category for probability theory?

What is the shape of the perfect coffee cup for heat retention assuming coffee is being drunk at a constant rate?

Representation of integers by $A = B^{\frac{5}{4}} + C^{\frac{5}{4}}$

Asymptotic behavior of $\int_{0}^r \frac{\sin(x)^2}{x^2} x^{\alpha}dx$.

Showing an integral is equal to arcsine

Matrix performing local differintegral analysis being its own inverse. Coincidence?

What is the value of $\lim\limits_{n\to\infty}2^{n}x_{n}$ if $x_1\in(0,1)$ and $x_{n+1}=\frac{\sqrt{1+x_n}-\sqrt{1-x_n}}2$ for every $n\ge1$?

Does there exist a real function with domain $\Bbb{R}$ such that $f'(x)>0$ and $f''(x)+(f'(x))^2<0$ for all $x$?

How I cut my orange - spherical volume integral

$\int_0^\infty \log\left(1 - \frac{x}{\sinh x}\right)\,dx$ and generalisations

Show that $\int_0^\pi\int_0^\pi\frac{\sin(x) \sin(px) \cos(qy)}{\sin(x)^2 + \sin(y)^2}\,dx\,dy$ tends to $0$ as $p\to\infty$ or $q\to\infty$

Calculating $\operatorname{cov}(\lambda\sinh(\frac{z-\gamma}{\delta})+\xi, \sigma z+\mu)$

Analogs of $\sum_{n\ge1}\frac{n^{13}}{e^{2\pi n}-1}=\frac{1}{24}$

Is it true that $(2^\ell-1)\sum_{k=1}^\ell\binom\ell{k}(\frac1n)^{2k}(1-\frac1n)^{2\ell-2k}+2(1-\frac1n)^\ell-1-(1-\frac1n)^{2\ell}\geq0$?

Prove $\log(x + 1) \leq x - \frac{x^{2}}{a}$ for fixed $a > 0$

Problem with Veblen's proof for the transcendence of $\pi$

The integral $\int_{0}^{\pi/2} \tan^p x~dx$

Can $\operatorname{Re}(a+bi)^{n}$ be overlapped with $a,b\in\mathbb{Z}$ fixed?

Is this a new characteristic function for the primes?

Find the smallest value of $\alpha\in \mathbb{R}$ such that for all $x>0$ you have $\left(1+\frac{1}{x}\right)^{x+\alpha}>e$

Rotation of Hyperbola with any angle

An integration to first order

Prove that $\int_0^\infty \frac{1+2\cos x+x\sin x}{1+2x\sin x +x^2}dx=\frac{\pi}{1+\Omega}$ where $\Omega e^\Omega=1$

Finding $\sum_{n=1}^{\infty}\frac{(-1)^n (H_{2n}-H_{n})}{n2^n \binom{2n}{n}}$

How to evaluate $\sum_{k=1}^{\infty}\frac{B\left(k, \frac{1}{2}\right)}{(2k+1)^2}$, where $B(x, y)$ is the Beta function?

How to solve $\sqrt{x+2}\geq x$?

Prove that $\int_0^\frac{\pi}{4}\frac{\cos (n-2)x}{\cos^nx}dx=\frac{1}{n-1}2^\frac{n-1}{2}\sin\frac{(n-1)\pi}{4}$

Number of zeros of $f(x)= \frac{1}{2} E\left[ \tanh \left( \frac{x+Z}{2} \right) \right]-\tanh(x)+\frac{x}{2}$ where $Z$ is standard normal

Can this formula for $\zeta(3)$ be proven or simplified further?