Question

Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$

score 1.8k · Answer 1 · 2014-07-25 08:48:44

1.8k

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Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$

Published on 25 Jul 2014 - 8:48

score 105 · Answer 2 · 2014-07-25 14:28:57

105

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Explain why $\big(\int_{-\infty}^{\infty}e^{-z^2/2}dz \big)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 + u^2)/2}dzdu$

Published on 25 Jul 2014 - 14:28

score 163 · Answer 3 · 2014-07-25 15:22:59

163

Views

Logarithmic integral and natural numbers.

Published on 25 Jul 2014 - 15:22

score 377 · Answer 4 · 2014-07-25 15:41:58

377

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Duo Fresnel-like integrals $(??)$

Published on 25 Jul 2014 - 15:41

score 339 · Answer 5 · 2014-07-25 23:14:33

339

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Evaluate $\int_0^1\frac{x^a-x^{-a}}{x-1}dx$

Published on 25 Jul 2014 - 23:14

score 1.2k · Answer 6 · 2026-03-27 21:44:37

1.2k

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Prove ${\large\int}_0^\infty\left({_2F_1}\left(\frac16,\frac12;\frac13;-x\right)\right)^{12}dx\stackrel{\color{#808080}?}=\frac{80663}{153090}$

Published on 27 Mar 2026 - 21:44

score 600 · Answer 7 · 2014-07-26 15:43:17

600

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An exercise from my brother: $\int_{-1}^1\frac{\ln (2x-1)}{\sqrt[\large 6]{x(1-x)(1-2x)^4}}\,dx$

Published on 26 Jul 2014 - 15:43

score 57 · Answer 8 · 2014-07-27 08:38:25

57

Views

Partial fraction for integrating

Published on 27 Jul 2014 - 8:38

score 262 · Answer 9 · 2014-07-27 17:34:20

262

Views

Value of the integral $\int_{\mathbb{R}} \frac{x\sin {(\pi x)}}{(1+x^2)^2}$

Published on 27 Jul 2014 - 17:34

score 204 · Answer 10 · 2014-07-28 06:47:06

204

Views

Show that $\int^\infty_0\left(\frac{\ln(1+x)} x\right)^2dx$ converge.

Published on 28 Jul 2014 - 6:47

score 572 · Answer 11 · 2014-07-28 11:21:46

572

Views

Evaluate $\int_0^\infty\frac{dl}{(r^2+l^2)^{\frac32}}$

Published on 28 Jul 2014 - 11:21

score 244 · Answer 12 · 2014-07-29 01:05:05

244

Views

Improper integral of $\frac{\ln x}x$

Published on 29 Jul 2014 - 1:05

score 213 · Answer 13 · 2014-07-29 06:37:39

213

Views

How to do integral $\int_0^{\infty} e^{-x^2-ax^4}\ dx , \ \text{ for $a>0$}$

Published on 29 Jul 2014 - 6:37

score 428 · Answer 14 · 2014-07-29 07:36:49

428

Views

Prove $\int_{\mathbb{R^{+}}} \frac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\frac{\pi}{32} (3\pi-4)^2$

Published on 29 Jul 2014 - 7:36

score 181 · Answer 15 · 2014-07-29 12:10:40

181

Views

Evaluate $\int_0^{+\infty } \frac{\log(t)}{1+t^2} \, dt$

Published on 29 Jul 2014 - 12:10

List Question

Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$

Explain why $\big(\int_{-\infty}^{\infty}e^{-z^2/2}dz \big)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 + u^2)/2}dzdu$

Logarithmic integral and natural numbers.

Duo Fresnel-like integrals $(??)$

Evaluate $\int_0^1\frac{x^a-x^{-a}}{x-1}dx$

Prove ${\large\int}_0^\infty\left({_2F_1}\left(\frac16,\frac12;\frac13;-x\right)\right)^{12}dx\stackrel{\color{#808080}?}=\frac{80663}{153090}$

An exercise from my brother: $\int_{-1}^1\frac{\ln (2x-1)}{\sqrt[\large 6]{x(1-x)(1-2x)^4}}\,dx$

Partial fraction for integrating

Value of the integral $\int_{\mathbb{R}} \frac{x\sin {(\pi x)}}{(1+x^2)^2}$

Show that $\int^\infty_0\left(\frac{\ln(1+x)} x\right)^2dx$ converge.

Evaluate $\int_0^\infty\frac{dl}{(r^2+l^2)^{\frac32}}$

Improper integral of $\frac{\ln x}x$

How to do integral $\int_0^{\infty} e^{-x^2-ax^4}\ dx , \ \text{ for $a>0$}$

Prove $\int_{\mathbb{R^{+}}} \frac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\frac{\pi}{32} (3\pi-4)^2$

Evaluate $\int_0^{+\infty } \frac{\log(t)}{1+t^2} \, dt$

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