Question

How can i find closed-form formula of this

score 66 · Answer 1 · 2024-01-02 19:06:57

66

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How can i find closed-form formula of this

Published on 02 Jan 2024 - 19:06

score 54 · Answer 2 · 2024-01-03 09:50:22

54

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Can you derive a closed form for this Fourier series: $\sum_{n=1}^{\infty} \frac{1}{n^2}\frac{a^n+b^n}{1-(ab)^n}\sin(n\phi)$

Published on 03 Jan 2024 - 9:50

score 223 · Answer 3 · 2024-01-03 21:15:43

223

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Show that $\int_{0}^{\frac{\pi}{2}} \log(1 + \tan^4 x) \cos^2 x \, dx = \frac{\pi \log(6 + 4\sqrt{2})}{4} - \frac{\pi}{2}$

Published on 03 Jan 2024 - 21:15

score 91 · Answer 4 · 2024-01-04 02:19:36

91

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Find the limit and integral $\lim_{\epsilon \to 0} \int_{\epsilon}^{1} \frac{x \sqrt{x} \log(x)}{x^4 + x^2 + 1} \, dx $

Published on 04 Jan 2024 - 2:19

score 124 · Answer 5 · 2024-01-04 05:53:58

124

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evaluate $\int_{0}^{1} \log\left(\frac{(\sqrt{1 - x} - 1)(\sqrt{1 - x^2} + 1)}{(\sqrt{1 - x} + 1)(\sqrt{1 - x^2} - 1)}\right) \arcsin(x^2) \, dx$

Published on 04 Jan 2024 - 5:53

score 1.6k · Answer 6 · 2024-01-04 09:19:54

1.6k

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Show that $\int_0^{\pi/3}\arccos(2\sin^2 x-\cos x)\mathrm dx=\frac{\pi^2}{5}$

Published on 04 Jan 2024 - 9:19

score 64 · Answer 7 · 2024-01-05 09:53:45

64

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Closed form for ${_3F_2}\!\left(\begin{array}c\tfrac34,1,1\\\tfrac32,\tfrac74\end{array}\middle|1\right)$

Published on 05 Jan 2024 - 9:53

score 91 · Answer 8 · 2024-01-06 08:15:47

91

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A challenging Integral Involving Logarithmic and Trigonometric Functions

Published on 06 Jan 2024 - 8:15

score 119 · Answer 9 · 2024-01-06 10:23:22

119

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Show that $\int_{0}^{1} \frac{\log(1 - x^2)}{\sqrt{x} (\sqrt{x} + 1)} \,dx = \frac{7}{2} \ln(2) - \frac{5}{4} \zeta(2)$

Published on 06 Jan 2024 - 10:23

score 181 · Answer 10 · 2024-01-07 19:33:48

181

Views

A closed form for integrals of the type $\Gamma\left(\sigma+it\right)$?

Published on 07 Jan 2024 - 19:33

score 552 · Answer 11 · 2026-03-25 20:13:49

552

Views

Simplify/re-express a potential expression for the inverse of $\frac{\sinh(x)}x$

Published on 25 Mar 2026 - 20:13

score 57 · Answer 12 · 2024-01-08 12:13:30

57

Views

Primitive of $e^\frac{-1}{1-x^2}$

Published on 08 Jan 2024 - 12:13

score 38 · Answer 13 · 2024-01-10 10:14:37

38

Views

Analytical Solution for a Double Integral Involving Logistic Functions and Gaussian Distributions

Published on 10 Jan 2024 - 10:14

score 500 · Answer 14 · 2026-03-28 00:49:22

500

Views

Closed form of integral $\int_{0}^{\infty}\frac{e^{-x}}{1+x/\cdots}dx$ with continued fraction

Published on 28 Mar 2026 - 0:49

score 360 · Answer 15 · 2024-01-12 14:38:31

360

Views

Show that $\int_0^1\int_{1-y}^1\sqrt{(x-1)(y-1)(x+y-1)}\mathrm dx\mathrm dy=\frac{2\pi}{105}$.

Published on 12 Jan 2024 - 14:38

List Question

How can i find closed-form formula of this

Can you derive a closed form for this Fourier series: $\sum_{n=1}^{\infty} \frac{1}{n^2}\frac{a^n+b^n}{1-(ab)^n}\sin(n\phi)$

Show that $\int_{0}^{\frac{\pi}{2}} \log(1 + \tan^4 x) \cos^2 x \, dx = \frac{\pi \log(6 + 4\sqrt{2})}{4} - \frac{\pi}{2}$

Find the limit and integral $\lim_{\epsilon \to 0} \int_{\epsilon}^{1} \frac{x \sqrt{x} \log(x)}{x^4 + x^2 + 1} \, dx $

evaluate $\int_{0}^{1} \log\left(\frac{(\sqrt{1 - x} - 1)(\sqrt{1 - x^2} + 1)}{(\sqrt{1 - x} + 1)(\sqrt{1 - x^2} - 1)}\right) \arcsin(x^2) \, dx$

Show that $\int_0^{\pi/3}\arccos(2\sin^2 x-\cos x)\mathrm dx=\frac{\pi^2}{5}$

Closed form for ${_3F_2}\!\left(\begin{array}c\tfrac34,1,1\\\tfrac32,\tfrac74\end{array}\middle|1\right)$

A challenging Integral Involving Logarithmic and Trigonometric Functions

Show that $\int_{0}^{1} \frac{\log(1 - x^2)}{\sqrt{x} (\sqrt{x} + 1)} \,dx = \frac{7}{2} \ln(2) - \frac{5}{4} \zeta(2)$

A closed form for integrals of the type $\Gamma\left(\sigma+it\right)$?

Simplify/re-express a potential expression for the inverse of $\frac{\sinh(x)}x$

Primitive of $e^\frac{-1}{1-x^2}$

Analytical Solution for a Double Integral Involving Logistic Functions and Gaussian Distributions

Closed form of integral $\int_{0}^{\infty}\frac{e^{-x}}{1+x/\cdots}dx$ with continued fraction

Show that $\int_0^1\int_{1-y}^1\sqrt{(x-1)(y-1)(x+y-1)}\mathrm dx\mathrm dy=\frac{2\pi}{105}$.

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