User Machinato

20 votes

Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods?

answered Jun 5, 2016 at 23:41

18 votes

Accepted

Sum the series $\sum_{n = 1}^{\infty}\{\coth (n\pi x) + x^{2}\coth(n\pi/x)\}/n^{3}$

answered Jul 22, 2016 at 12:09

16 votes

The Basel problem

answered Jun 8, 2017 at 23:09

13 votes

Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$

answered Mar 2 at 10:17

10 votes

The Basel problem

answered Jun 20, 2016 at 21:18

9 votes

Evaluation of Gaussian integral $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx$

answered Apr 26, 2016 at 16:43

9 votes

Proof of $\sum_{n=1}^{\infty}\frac1{n^3}\frac{\sinh\pi n\sqrt2-\sin\pi n\sqrt2}{{\cosh\pi n\sqrt2}-\cos\pi n\sqrt2}=\frac{\pi^3}{18\sqrt2}$

answered Jul 22, 2016 at 2:21

8 votes

The Basel problem

answered Jun 21, 2016 at 0:06

8 votes

Calculate $\sum\limits_{n=1}^{\infty} \frac{a\cos(nx)}{a^2+n^2}$

answered Jun 29, 2016 at 10:53

8 votes

Closed form for $\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}$ conjectured

answered Aug 3, 2016 at 12:32

5 votes

Accepted

Finding the geometric shape

answered Jun 7, 2017 at 17:52

5 votes

How was the normal distribution derived?

answered Jan 23, 2020 at 23:43

4 votes

Conjectured closed form for $\sum_{n=-\infty}^\infty\frac{1}{\cosh\pi n+\frac{1}{\sqrt{2}}}$

answered Jun 12, 2016 at 23:38

3 votes

Closed form for $\sum_{n=-\infty}^{\infty} \frac{1}{n^4+a^4}$

answered Jun 17, 2016 at 22:58

3 votes

Integrate $\int \sqrt{\frac{x}{x+1}}dx$

answered Jun 3, 2016 at 19:30

3 votes

Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$?

answered Jan 13, 2022 at 12:50

3 votes

Accepted

Average distance between two points lying on boundary of a square

answered Jul 18, 2016 at 12:00

3 votes

Accepted

Compute the sum of the series.

answered Jul 19, 2016 at 1:17

3 votes

Deriving Maclaurin series for $\frac{\arcsin x}{\sqrt{1-x^2}}$.

answered May 18, 2018 at 16:12

3 votes

Does the number of cycles in a random permutation converge to Poisson as $n\to\infty$?

answered Jan 30, 2019 at 10:26

3 votes

Integral $\int_0^\infty e^{-x^2} \cos\alpha x \ dx$

answered Feb 17, 2018 at 19:55

3 votes

Accepted

Evaluating double integral $\int_0^\infty\int_{-1}^1 r^{n-1} e^{-\alpha^2(r^2 - r x)} (1-x^2)^{\frac{n-3}{2}} \, \mathrm{d}x\, \mathrm{d}r$

answered Oct 7, 2020 at 10:21

3 votes

Asymptotic expansion of a coefficients given functional equation of the generating function

answered Dec 21, 2022 at 22:56

2 votes

Accepted

Series of product of legendre polynomials with shifted degree

answered May 14 at 16:50

2 votes

Integration validity of $\int\frac{1}{\sqrt{a^2 + x^2}}\,dx$

answered Jun 4, 2016 at 16:13

2 votes

Elliptic definite integral $\int_0^1 x \left(\left(7+x^2\right) E\left(x\right) - 4 \left(1-x^2\right) K(x)\right)^2 \mathrm{d}x$

answered Nov 14, 2020 at 19:04

2 votes

The Postman Problem

answered Feb 15, 2018 at 0:53

2 votes

Mean distance between 2 points in a ball

answered Aug 27, 2020 at 20:34

2 votes

How to integrate $\int\frac{\sqrt{1-x}}{\sqrt{x}}\ \mathrm dx$

answered Jun 4, 2016 at 16:52

1 vote

Integral $\int\sqrt{\sin2x}\operatorname d\!x$

answered Jun 5, 2016 at 18:22

Stack Exchange Network

50 Answers

Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods?

Sum the series $\sum_{n = 1}^{\infty}\{\coth (n\pi x) + x^{2}\coth(n\pi/x)\}/n^{3}$

The Basel problem

Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$

The Basel problem

Evaluation of Gaussian integral $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx$

Proof of $\sum_{n=1}^{\infty}\frac1{n^3}\frac{\sinh\pi n\sqrt2-\sin\pi n\sqrt2}{{\cosh\pi n\sqrt2}-\cos\pi n\sqrt2}=\frac{\pi^3}{18\sqrt2}$

The Basel problem

Calculate $\sum\limits_{n=1}^{\infty} \frac{a\cos(nx)}{a^2+n^2}$

Closed form for $\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}$ conjectured

Finding the geometric shape

How was the normal distribution derived?

Conjectured closed form for $\sum_{n=-\infty}^\infty\frac{1}{\cosh\pi n+\frac{1}{\sqrt{2}}}$

Closed form for $\sum_{n=-\infty}^{\infty} \frac{1}{n^4+a^4}$

Integrate $\int \sqrt{\frac{x}{x+1}}dx$

Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$?

Average distance between two points lying on boundary of a square

Compute the sum of the series.

Deriving Maclaurin series for $\frac{\arcsin x}{\sqrt{1-x^2}}$.

Does the number of cycles in a random permutation converge to Poisson as $n\to\infty$?

Integral $\int_0^\infty e^{-x^2} \cos\alpha x \ dx$

Evaluating double integral $\int_0^\infty\int_{-1}^1 r^{n-1} e^{-\alpha^2(r^2 - r x)} (1-x^2)^{\frac{n-3}{2}} \, \mathrm{d}x\, \mathrm{d}r$

Asymptotic expansion of a coefficients given functional equation of the generating function

Series of product of legendre polynomials with shifted degree

Integration validity of $\int\frac{1}{\sqrt{a^2 + x^2}}\,dx$

Elliptic definite integral $\int_0^1 x \left(\left(7+x^2\right) E\left(x\right) - 4 \left(1-x^2\right) K(x)\right)^2 \mathrm{d}x$

The Postman Problem

Mean distance between 2 points in a ball

How to integrate $\int\frac{\sqrt{1-x}}{\sqrt{x}}\ \mathrm dx$

Integral $\int\sqrt{\sin2x}\operatorname d\!x$