Question

Prove that $\int_0^1\frac{\ln(1-x)\ln^2x}{x-1}dx=\frac{\pi^4}{180}$

score 1k · Answer 1 · 2026-03-26 12:49:20

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Prove that $\int_0^1\frac{\ln(1-x)\ln^2x}{x-1}dx=\frac{\pi^4}{180}$

Published on 26 Mar 2026 - 12:49

score 490 · Answer 2 · 2014-08-06 14:27:15

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Prove that $\int_0^1\frac{1-x}{1-x^6}\ln^4x\,dx=\frac{16\sqrt{3}}{729}\pi^5+\frac{605}{54}\zeta(5)$

Published on 06 Aug 2014 - 14:27

score 45 · Answer 3 · 2014-08-06 23:14:41

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Show the convergence of the integral $F(t,x)=\int_{-\infty}^{+\infty}\exp[i\tau t-(i\tau)^{1/2}x - (i\tau)^\theta] \,d\tau$

Published on 06 Aug 2014 - 23:14

score 1.7k · Answer 4 · 2014-08-07 16:40:53

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Proof of $\int_0^\infty \frac{x^{\alpha}dx}{1+2x\cos\beta +x^{2}}=\frac{\pi\sin (\alpha\beta)}{\sin (\alpha\pi)\sin \beta }$

Published on 07 Aug 2014 - 16:40

score 1k · Answer 5 · 2026-03-29 04:47:20

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Integral $\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x=0$

Published on 29 Mar 2026 - 4:47

score 885 · Answer 6 · 2014-08-08 08:53:21

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The other ways to calculate $\int_0^1\frac{\ln(1-x^2)}{x}dx$

Published on 08 Aug 2014 - 8:53

score 120 · Answer 7 · 2014-08-09 06:03:16

120

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Evaluate $\frac{1}{a}\int_0^\infty{x^2}e^{-\frac{x^2}{2a}}\,dx$

Published on 09 Aug 2014 - 6:03

score 65.3k · Answer 8 · 2014-08-09 08:00:22

65.3k

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Integration of the cardinal sine

Published on 09 Aug 2014 - 8:00

score 83 · Answer 9 · 2014-08-10 01:09:19

What is the relation between improper integrals $\int_{-\infty}^{\infty}xf(x)dx$ and $\int_{0}^{\infty}xf(x)dx-\int_{-\infty}^{0}|x|f(x)dx$?

score 397 · Answer 10 · 2014-08-10 03:21:18

397

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Integral of $\int_0^\infty\frac{1}{e^{x}-x} dx$

Published on 10 Aug 2014 - 3:21

score 320 · Answer 11 · 2026-03-28 06:40:50

320

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integral over product of two bessel functions at discontinuity

Published on 28 Mar 2026 - 6:40

score 90 · Answer 12 · 2014-08-11 14:35:45

90

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How to evaluate limits having infinity by infinity form

Published on 11 Aug 2014 - 14:35

score 824 · Answer 13 · 2014-08-12 02:13:29

824

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Is this proof correct? Divergence of $\int_{1}^{\infty} \left| \frac{\sin x}{x} \right| \, \mathrm{d}x $

Published on 12 Aug 2014 - 2:13

score 8k · Answer 14 · 2014-08-12 02:39:44

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How to recognize and evaluate improper integrals when the interval of integration is finite?

Published on 12 Aug 2014 - 2:39

score 2.6k · Answer 15 · 2026-03-28 10:34:49

2.6k

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Closed form for integral of integer powers of Sinc function

Published on 28 Mar 2026 - 10:34

List Question

Prove that $\int_0^1\frac{\ln(1-x)\ln^2x}{x-1}dx=\frac{\pi^4}{180}$

Prove that $\int_0^1\frac{1-x}{1-x^6}\ln^4x\,dx=\frac{16\sqrt{3}}{729}\pi^5+\frac{605}{54}\zeta(5)$

Show the convergence of the integral $F(t,x)=\int_{-\infty}^{+\infty}\exp[i\tau t-(i\tau)^{1/2}x - (i\tau)^\theta] \,d\tau$

Proof of $\int_0^\infty \frac{x^{\alpha}dx}{1+2x\cos\beta +x^{2}}=\frac{\pi\sin (\alpha\beta)}{\sin (\alpha\pi)\sin \beta }$

Integral $\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x=0$

The other ways to calculate $\int_0^1\frac{\ln(1-x^2)}{x}dx$

Evaluate $\frac{1}{a}\int_0^\infty{x^2}e^{-\frac{x^2}{2a}}\,dx$

Integration of the cardinal sine

What is the relation between improper integrals $\int_{-\infty}^{\infty}xf(x)dx$ and $\int_{0}^{\infty}xf(x)dx-\int_{-\infty}^{0}|x|f(x)dx$?

Integral of $\int_0^\infty\frac{1}{e^{x}-x} dx$

integral over product of two bessel functions at discontinuity

How to evaluate limits having infinity by infinity form

Is this proof correct? Divergence of $\int_{1}^{\infty} \left| \frac{\sin x}{x} \right| \, \mathrm{d}x $

How to recognize and evaluate improper integrals when the interval of integration is finite?

Closed form for integral of integer powers of Sinc function

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