Question

A property of a kind of product integral

score 43 · Answer 1 · 2020-06-14 17:17:53

43

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A property of a kind of product integral

Published on 14 Jun 2020 - 17:17

score 186 · Answer 2 · 2020-06-15 09:50:11

186

Views

Does $\lim \limits_{n\to\infty} \int_0^1 \sin(\frac{1}{x}) \sin(nx)dx$ exist?

Published on 15 Jun 2020 - 9:50

score 106 · Answer 3 · 2026-04-01 22:34:49

106

Views

How do you prove that $\ln(x) = \int_0^\infty \frac{e^{-t}-e^{-xt}}{t}$?

Published on 01 Apr 2026 - 22:34

score 238 · Answer 4 · 2026-04-06 07:44:23

238

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Evaluate $\int_0^1 \ln{\left(\Gamma(x)\right)}\cos^2{(\pi x)} \; {\mathrm{d}x}$

Published on 06 Apr 2026 - 7:44

score 63 · Answer 5 · 2020-06-16 14:19:59

63

Views

Prove there exists $\alpha \ge 0$ s.t $\int_0^\alpha f(x)dx =\int_0^\infty g(x)dx$ given that $f,g\ge 0$, $F(x)$ diverges and $G(x)$ converges

Published on 16 Jun 2020 - 14:19

score 80 · Answer 6 · 2020-06-17 12:22:02

80

Views

Convergence of $\int_0^\infty \frac{\ln(1+x^{-2a})}{\sqrt{x^a+x^{-a}}}dx$ without Gamma function

Published on 17 Jun 2020 - 12:22

score 714 · Answer 7 · 2020-06-17 13:04:07

714

Views

find the real values of p and q such that following integral converges

Published on 17 Jun 2020 - 13:04

score 79 · Answer 8 · 2020-06-17 22:14:01

79

Views

Integration of $\frac{1}{u^4 + (4\zeta^2-2)u^2 + 1}$

Published on 17 Jun 2020 - 22:14

score 83 · Answer 9 · 2020-06-18 06:27:12

83

Views

What are the values of $a$ for which this integral converges?

Published on 18 Jun 2020 - 6:27

score 81 · Answer 10 · 2026-03-26 02:57:40

81

Views

Prove the convergence of the integral $\int_{0}^{\infty}\frac{x^n}{\Gamma\Big(\operatorname{W}(x)\Big)}dx$

Published on 26 Mar 2026 - 2:57

score 202 · Answer 11 · 2020-06-18 15:28:02

202

Views

$f(c)\int_{1}^{\infty} f(x) \phi(x) dx =g(c)\int_{1}^{\infty} g(x) \phi(x) dx$

Published on 18 Jun 2020 - 15:28

score 203 · Answer 12 · 2026-03-27 13:20:26

203

Views

What 1D functions y=f(x) become rotationally symmetric as z=f(x)f(y)?

Published on 27 Mar 2026 - 13:20

score 186 · Answer 13 · 2020-06-19 11:53:39

186

Views

Does the following integral converge? $ \int\limits_0^\pi\frac{\sin x}{\sqrt{x}}\ dx $

Published on 19 Jun 2020 - 11:53

score 269 · Answer 14 · 2020-06-19 19:54:23

269

Views

Evaluating Improper Integral With No Closed-Form Antiderivative

Published on 19 Jun 2020 - 19:54

score 779 · Answer 15 · 2020-06-19 21:36:10

779

Views

Prove: $\int_{0}^{\pi/2}{\sqrt{1 + \sqrt{1 + (\tan{x})^{2/3}}}\,dx} = \frac{\pi}{2} (3^{1/4} + 3^{3/4} - 2)$

Published on 19 Jun 2020 - 21:36

List Question

A property of a kind of product integral

Does $\lim \limits_{n\to\infty} \int_0^1 \sin(\frac{1}{x}) \sin(nx)dx$ exist?

How do you prove that $\ln(x) = \int_0^\infty \frac{e^{-t}-e^{-xt}}{t}$?

Evaluate $\int_0^1 \ln{\left(\Gamma(x)\right)}\cos^2{(\pi x)} \; {\mathrm{d}x}$

Prove there exists $\alpha \ge 0$ s.t $\int_0^\alpha f(x)dx =\int_0^\infty g(x)dx$ given that $f,g\ge 0$, $F(x)$ diverges and $G(x)$ converges

Convergence of $\int_0^\infty \frac{\ln(1+x^{-2a})}{\sqrt{x^a+x^{-a}}}dx$ without Gamma function

find the real values of p and q such that following integral converges

Integration of $\frac{1}{u^4 + (4\zeta^2-2)u^2 + 1}$

What are the values of $a$ for which this integral converges?

Prove the convergence of the integral $\int_{0}^{\infty}\frac{x^n}{\Gamma\Big(\operatorname{W}(x)\Big)}dx$

$f(c)\int_{1}^{\infty} f(x) \phi(x) dx =g(c)\int_{1}^{\infty} g(x) \phi(x) dx$

What 1D functions y=f(x) become rotationally symmetric as z=f(x)f(y)?

Does the following integral converge? $ \int\limits_0^\pi\frac{\sin x}{\sqrt{x}}\ dx $

Evaluating Improper Integral With No Closed-Form Antiderivative

Prove: $\int_{0}^{\pi/2}{\sqrt{1 + \sqrt{1 + (\tan{x})^{2/3}}}\,dx} = \frac{\pi}{2} (3^{1/4} + 3^{3/4} - 2)$

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