Question

Integral $\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx$

score 8.4k · Answer 1 · 2013-11-17 21:42:15

8.4k

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Integral $\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx$

Published on 17 Nov 2013 - 21:42

score 105 · Answer 2 · 2013-11-17 22:26:43

105

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How can I compute this sum of binomial

Published on 17 Nov 2013 - 22:26

score 529 · Answer 3 · 2013-11-18 06:07:51

529

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closed form for $\int_0^{\infty}\log^n\left(\frac{e^x}{e^x-1}\right)dx$

Published on 18 Nov 2013 - 6:07

score 250 · Answer 4 · 2013-11-18 19:25:08

250

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Derivatives of the Struve functions $H_\nu(x)$, $L_\nu(x)$ and other related functions w.r.t. their index $\nu$

Published on 18 Nov 2013 - 19:25

score 2.1k · Answer 5 · 2013-11-19 06:02:57

2.1k

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Closed form for $\int_0^\infty\frac{\sin x\,\cdot\,\operatorname{Ci}x-\cos x\,\cdot\,\operatorname{Si}x}{\sqrt{16\,x^2+1}}dx$

Published on 19 Nov 2013 - 6:02

score 846 · Answer 6 · 2013-11-20 18:39:44

846

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Simplification of $G_{2,4}^{4,2}\left(\frac18,\frac12\middle|\begin{array}{c}\frac12,\frac12\\0,0,\frac12,\frac12\\\end{array}\right)$

Published on 20 Nov 2013 - 18:39

score 112 · Answer 7 · 2013-11-21 17:44:45

112

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The closed form of a sum of mod(k,m) where k goes from 1 to a arbitrary number.

Published on 21 Nov 2013 - 17:44

score 1.6k · Answer 8 · 2013-11-21 19:00:45

1.6k

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A closed form for $\int_0^1{_2F_1}\left(-\frac{1}{4},\frac{5}{4};\,1;\,\frac{x}{2}\right)^2dx$

Published on 21 Nov 2013 - 19:00

score 1k · Answer 9 · 2013-11-22 06:48:18

1k

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Integral $\int_0^\infty{_1F_2}\left(\begin{array}{c}\tfrac12\\1,\tfrac32\end{array}\middle|-x\right)\frac{dx}{1+4\,x}$

Published on 22 Nov 2013 - 6:48

score 572 · Answer 10 · 2013-11-22 20:30:16

572

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How to prove $\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$?

Published on 22 Nov 2013 - 20:30

score 3.4k · Answer 11 · 2013-11-22 23:55:46

3.4k

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Integral $\int_0^\infty\frac{\ln\left(1+x+\sqrt{x^2+2\,x}\right)\,\ln\left(1+\sqrt{x^2+2\,x+2}\right)}{x^2+2x+1}dx$

Published on 22 Nov 2013 - 23:55

score 1.6k · Answer 12 · 2013-11-23 04:32:56

1.6k

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Derivative of the Meijer G-function with respect to one of its parameters

Published on 23 Nov 2013 - 4:32

score 2.1k · Answer 13 · 2013-11-23 22:33:15

2.1k

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Integral $\int_0^\infty\frac{\ln\left(\sqrt{x+1\vphantom{x^0}}-1\right)\,\ln\left(\sqrt{x^{-1}+1}+1\right)}{(x+1)^{3/2}}dx$

Published on 23 Nov 2013 - 22:33

score 290 · Answer 14 · 2013-11-25 03:04:36

290

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Closed form for integral $ \int_0^{\pi} \frac{\sin (m \phi)}{(1 + r \cos \phi)^n} d\phi$

Published on 25 Nov 2013 - 3:04

score 651 · Answer 15 · 2013-11-25 03:39:52

651

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Integral $S_\ell(r) = \int_0^{\pi}\int_{\phi}^{\pi}\frac{(1+ r \cos \psi)^{\ell+1}}{(1+ r \cos \phi)^\ell} \rm d\psi \ \rm d\phi $

Published on 25 Nov 2013 - 3:39

List Question

Integral $\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx$

How can I compute this sum of binomial

closed form for $\int_0^{\infty}\log^n\left(\frac{e^x}{e^x-1}\right)dx$

Derivatives of the Struve functions $H_\nu(x)$, $L_\nu(x)$ and other related functions w.r.t. their index $\nu$

Closed form for $\int_0^\infty\frac{\sin x\,\cdot\,\operatorname{Ci}x-\cos x\,\cdot\,\operatorname{Si}x}{\sqrt{16\,x^2+1}}dx$

Simplification of $G_{2,4}^{4,2}\left(\frac18,\frac12\middle|\begin{array}{c}\frac12,\frac12\\0,0,\frac12,\frac12\\\end{array}\right)$

The closed form of a sum of mod(k,m) where k goes from 1 to a arbitrary number.

A closed form for $\int_0^1{_2F_1}\left(-\frac{1}{4},\frac{5}{4};\,1;\,\frac{x}{2}\right)^2dx$

Integral $\int_0^\infty{_1F_2}\left(\begin{array}{c}\tfrac12\\1,\tfrac32\end{array}\middle|-x\right)\frac{dx}{1+4\,x}$

How to prove $\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$?

Integral $\int_0^\infty\frac{\ln\left(1+x+\sqrt{x^2+2\,x}\right)\,\ln\left(1+\sqrt{x^2+2\,x+2}\right)}{x^2+2x+1}dx$

Derivative of the Meijer G-function with respect to one of its parameters

Integral $\int_0^\infty\frac{\ln\left(\sqrt{x+1\vphantom{x^0}}-1\right)\,\ln\left(\sqrt{x^{-1}+1}+1\right)}{(x+1)^{3/2}}dx$

Closed form for integral $ \int_0^{\pi} \frac{\sin (m \phi)}{(1 + r \cos \phi)^n} d\phi$

Integral $S_\ell(r) = \int_0^{\pi}\int_{\phi}^{\pi}\frac{(1+ r \cos \psi)^{\ell+1}}{(1+ r \cos \phi)^\ell} \rm d\psi \ \rm d\phi $

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