Question

how to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?

score 178 · Answer 1 · 2026-03-27 12:28:59

178

Views

how to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?

Published on 27 Mar 2026 - 12:28

score 208 · Answer 2 · 2026-03-30 06:12:41

208

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Evaluate $\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{1}{\sqrt{3-x^2-y^2-z^2} }\text{d}x \text{d}y\text{d}z$

Published on 30 Mar 2026 - 6:12

score 253 · Answer 3 · 2024-02-15 18:56:35

253

Views

How to integrate $\int_{0}^{1} \int_{0}^{1} \tanh^{-1}\left(\frac{x}{y} + \frac{y}{x}\right) \,dx\,dy$

Published on 15 Feb 2024 - 18:56

score 184 · Answer 4 · 2024-02-16 11:39:40

184

Views

How to evaluate this sum $\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$

Published on 16 Feb 2024 - 11:39

score 47 · Answer 5 · 2026-03-31 19:47:51

47

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approximate a sum

Published on 31 Mar 2026 - 19:47

score 529 · Answer 6 · 2024-02-16 18:08:52

529

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How to evaluate $\int_{0}^{\frac{\pi}{2}} \frac{\cos(x)}{(1 + \sqrt{\sin(2x)})^n} \,dx$

Published on 16 Feb 2024 - 18:08

score 56 · Answer 7 · 2026-02-23 06:40:54

56

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What is the 1-case closed form for $\sum_{i = 1}^{x} \lfloor \frac{i - r}{d}\rfloor$?

Published on 23 Feb 2026 - 6:40

score 181 · Answer 8 · 2026-03-26 17:42:30

181

Views

What is the number of integers divisible by either 2, 3, or 5 from the integers 1 to $n+1$?

Published on 26 Mar 2026 - 17:42

score 73 · Answer 9 · 2024-02-17 19:17:13

73

Views

how to evaluate $\int_{0}^{1} \int_{0}^{1} \frac{\ln(1 - xy) \cdot \text{Li}_{4}(1 - x)}{x(1 - x)(1 - xy)} \,dy\,dx$

Published on 17 Feb 2024 - 19:17

score 120 · Answer 10 · 2024-02-18 15:39:35

120

Views

How to evaluate $\int_0^1 \dfrac{\operatorname{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\dfrac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$

Published on 18 Feb 2024 - 15:39

score 126 · Answer 11 · 2024-02-18 16:46:37

126

Views

How to evaluate $\int_0^{\infty } \left(\frac{1}{(x+1)^2 \log (x+1)}-\frac{\log (x+1) \tan ^{-1}(x)}{x^3}\right) \, dx$

Published on 18 Feb 2024 - 16:46

score 102 · Answer 12 · 2024-02-19 09:55:07

102

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Show that $\int_0^\infty \frac{x^2+1}{x^4+x^2+1}\frac{\log(1-x+x^2)}{\log(x)}dx=\frac{\pi}{\sqrt{3}}$

Published on 19 Feb 2024 - 9:55

score 169 · Answer 13 · 2024-02-19 11:36:51

169

Views

How to integrate $\int_0^{\pi/2} \frac{x(1+\sin^2 x)\cos x}{(3+\sin^2 x)(1+3\sin^2 x)}\,dx$

Published on 19 Feb 2024 - 11:36

score 117 · Answer 14 · 2024-02-20 12:24:43

117

Views

Calculation of $\sum_{n=1}^{\infty}\frac{\psi_1(n)}{2^nn^2}$

Published on 20 Feb 2024 - 12:24

score 248 · Answer 15 · 2024-02-21 11:45:01

248

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Show that $\int_{-\infty}^\infty \frac{e^x}{e^{2x}+e^{2a}}\frac{1}{x^2+\pi^2}dx = \frac{2\pi e^{-a}}{4a^2+\pi^2}-\frac{1}{1+e^{2a}}$

Published on 21 Feb 2024 - 11:45

List Question

how to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?

Evaluate $\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{1}{\sqrt{3-x^2-y^2-z^2} }\text{d}x \text{d}y\text{d}z$

How to integrate $\int_{0}^{1} \int_{0}^{1} \tanh^{-1}\left(\frac{x}{y} + \frac{y}{x}\right) \,dx\,dy$

How to evaluate this sum $\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$

approximate a sum

How to evaluate $\int_{0}^{\frac{\pi}{2}} \frac{\cos(x)}{(1 + \sqrt{\sin(2x)})^n} \,dx$

What is the 1-case closed form for $\sum_{i = 1}^{x} \lfloor \frac{i - r}{d}\rfloor$?

What is the number of integers divisible by either 2, 3, or 5 from the integers 1 to $n+1$?

how to evaluate $\int_{0}^{1} \int_{0}^{1} \frac{\ln(1 - xy) \cdot \text{Li}_{4}(1 - x)}{x(1 - x)(1 - xy)} \,dy\,dx$

How to evaluate $\int_0^1 \dfrac{\operatorname{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\dfrac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$

How to evaluate $\int_0^{\infty } \left(\frac{1}{(x+1)^2 \log (x+1)}-\frac{\log (x+1) \tan ^{-1}(x)}{x^3}\right) \, dx$

Show that $\int_0^\infty \frac{x^2+1}{x^4+x^2+1}\frac{\log(1-x+x^2)}{\log(x)}dx=\frac{\pi}{\sqrt{3}}$

How to integrate $\int_0^{\pi/2} \frac{x(1+\sin^2 x)\cos x}{(3+\sin^2 x)(1+3\sin^2 x)}\,dx$

Calculation of $\sum_{n=1}^{\infty}\frac{\psi_1(n)}{2^nn^2}$

Show that $\int_{-\infty}^\infty \frac{e^x}{e^{2x}+e^{2a}}\frac{1}{x^2+\pi^2}dx = \frac{2\pi e^{-a}}{4a^2+\pi^2}-\frac{1}{1+e^{2a}}$

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