Question

Calculate $\int_{-\infty}^{\infty}{\ \frac{dx}{\sqrt{1 + x^{4}}}\ x\ \sin [n\pi\phi(x)] \ }$; $\phi(x)\propto\int_{0}^{x}\frac{dy}{\sqrt{1+ y^{4}}} $

score 227 · Answer 1 · 2026-03-27 22:57:36

227

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Calculate $\int_{-\infty}^{\infty}{\ \frac{dx}{\sqrt{1 + x^{4}}}\ x\ \sin [n\pi\phi(x)] \ }$; $\phi(x)\propto\int_{0}^{x}\frac{dy}{\sqrt{1+ y^{4}}} $

Published on 27 Mar 2026 - 22:57

score 139 · Answer 2 · 2022-02-20 02:59:07

139

Views

Evaluation of $\int_{0}^{\frac{\pi}{2}} \frac{dx}{\sqrt[3]{\frac{1}{3} + \sin^2 x}} $

Published on 20 Feb 2022 - 2:59

score 108 · Answer 3 · 2022-02-21 04:00:56

108

Views

Hyperelliptic integral: Piecewise solution (handbook) vs General solution (Wolfram)

Published on 21 Feb 2022 - 4:00

score 217 · Answer 4 · 2026-03-26 17:43:22

217

Views

Is this function an alternative solution to the nonlinear pendulum?

Published on 26 Mar 2026 - 17:43

score 508 · Answer 5 · 2022-03-06 05:00:40

508

Views

Evaluate $\int_{0}^{1}K(x)^2\text{d}x -\int_{0}^{1} \frac{x\sqrt{1-x^2} }{2-x^2}K(x)^2\text{d}x$

Published on 06 Mar 2022 - 5:00

score 168 · Answer 6 · 2022-03-09 22:10:43

168

Views

Proving $\frac{\pi}{2}=\sum^\infty_{l=0} \frac{(-1)^l}{2l+1}\big(P_{2l}(x)+\text{sgn}(x)P_{2l+1}(x)\big)$

Published on 09 Mar 2022 - 22:10

score 284 · Answer 7 · 2026-03-27 17:57:38

284

Views

Computing : $\displaystyle \int \frac {m + n\cos (x-x_0) }{ [a+b\cos (x-x_0)]^{3/2} } \mathrm{d}x $ with integration of $E$ and $F$

Published on 27 Mar 2026 - 17:57

score 65 · Answer 8 · 2022-03-30 00:49:03

65

Views

Solving $ \int \sqrt{4\sin^2(x) + \cos^2(x) } dx $ without an elliptical integral

Published on 30 Mar 2022 - 0:49

score 261 · Answer 9 · 2026-03-27 23:01:27

261

Views

Closed form for definite integrals invovling Jacobi elliptic functions

Published on 27 Mar 2026 - 23:01

score 60 · Answer 10 · 2026-03-27 23:01:32

60

Views

Birationally equivalent elliptic curves and singularities

Published on 27 Mar 2026 - 23:01

score 151 · Answer 11 · 2022-04-23 16:37:38

151

Views

Is it possible to compute $\int_0^{\pi/2}\sqrt{a^2 \sin^2 x + b^2 \cos^2 x}\ dx$.

Published on 23 Apr 2022 - 16:37

score 57 · Answer 12 · 2022-04-25 14:48:49

57

Views

Show that $\int_0^{\pi/2} \frac{dt}{\sqrt{\sin t}}=\int_0^1 \frac{dx}{\sqrt{x-x^3}}$

Published on 25 Apr 2022 - 14:48

score 274 · Answer 13 · 2022-05-02 01:30:35

274

Views

Does the Incomplete Beta function have forms of Elliptic E besides $\frac14 \text B_{\sin^2(2x)}\left(\frac12,\frac34\right)=\text E(x,2)$?

Published on 02 May 2022 - 1:30

score 51 · Answer 14 · 2022-05-09 21:44:34

51

Views

Defnite integral of the square root of a fourth degree polynomiae

Published on 09 May 2022 - 21:44

score 125 · Answer 15 · 2022-05-21 05:16:23

125

Views

Prove that $\int_{0}^{1} x K(x) E(x) \,\mathrm{d} x{=}\frac{9}{16}+\frac{21}{32} \zeta(3)$

Published on 21 May 2022 - 5:16

List Question

Calculate $\int_{-\infty}^{\infty}{\ \frac{dx}{\sqrt{1 + x^{4}}}\ x\ \sin [n\pi\phi(x)] \ }$; $\phi(x)\propto\int_{0}^{x}\frac{dy}{\sqrt{1+ y^{4}}} $

Evaluation of $\int_{0}^{\frac{\pi}{2}} \frac{dx}{\sqrt[3]{\frac{1}{3} + \sin^2 x}} $

Hyperelliptic integral: Piecewise solution (handbook) vs General solution (Wolfram)

Is this function an alternative solution to the nonlinear pendulum?

Evaluate $\int_{0}^{1}K(x)^2\text{d}x -\int_{0}^{1} \frac{x\sqrt{1-x^2} }{2-x^2}K(x)^2\text{d}x$

Proving $\frac{\pi}{2}=\sum^\infty_{l=0} \frac{(-1)^l}{2l+1}\big(P_{2l}(x)+\text{sgn}(x)P_{2l+1}(x)\big)$

Computing : $\displaystyle \int \frac {m + n\cos (x-x_0) }{ [a+b\cos (x-x_0)]^{3/2} } \mathrm{d}x $ with integration of $E$ and $F$

Solving $ \int \sqrt{4\sin^2(x) + \cos^2(x) } dx $ without an elliptical integral

Closed form for definite integrals invovling Jacobi elliptic functions

Birationally equivalent elliptic curves and singularities

Is it possible to compute $\int_0^{\pi/2}\sqrt{a^2 \sin^2 x + b^2 \cos^2 x}\ dx$.

Show that $\int_0^{\pi/2} \frac{dt}{\sqrt{\sin t}}=\int_0^1 \frac{dx}{\sqrt{x-x^3}}$

Does the Incomplete Beta function have forms of Elliptic E besides $\frac14 \text B_{\sin^2(2x)}\left(\frac12,\frac34\right)=\text E(x,2)$?

Defnite integral of the square root of a fourth degree polynomiae

Prove that $\int_{0}^{1} x K(x) E(x) \,\mathrm{d} x{=}\frac{9}{16}+\frac{21}{32} \zeta(3)$

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