Question

How to evaluate $\int_0^\infty \frac{x^5(e^{3x}-e^x)}{(e^x-1)^4}\,\mathrm{d}x$

score 69 · Answer 1 · 2020-04-20 10:49:58

69

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How to evaluate $\int_0^\infty \frac{x^5(e^{3x}-e^x)}{(e^x-1)^4}\,\mathrm{d}x$

Published on 20 Apr 2020 - 10:49

score 381 · Answer 2 · 2026-03-28 09:33:35

381

Views

Value of $\sum_{n=1}^{\infty}\frac{1}{\sqrt{n!}}$?

Published on 28 Mar 2026 - 9:33

score 194 · Answer 3 · 2020-04-22 19:55:16

194

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Asymptotic Behaviour of $\Gamma^{(k)}(1)$

Published on 22 Apr 2020 - 19:55

score 106 · Answer 4 · 2026-03-25 09:48:19

106

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Branch of incomplete gamma function

Published on 25 Mar 2026 - 9:48

score 33 · Answer 5 · 2026-03-25 15:49:54

33

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What material do I need to cover to understand the gamma function?

Published on 25 Mar 2026 - 15:49

score 318 · Answer 6 · 2020-04-24 20:32:01

318

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Gamma Function Specifics and properties

Published on 24 Apr 2020 - 20:32

score 187 · Answer 7 · 2020-04-27 04:27:34

187

Views

How to prove $\lim_{n\to\infty}\frac{1}{\Gamma(n/2+1)}\int_{0}^{n} t^{n/2}e^{-t}dt = 1$?

Published on 27 Apr 2020 - 4:27

score 575 · Answer 8 · 2020-04-28 02:15:37

575

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Show $ \int_0^\infty \ x^{\alpha-1}e^{-x/\beta} \ dx = \beta^\alpha\Gamma(\alpha).$

Published on 28 Apr 2020 - 2:15

score 68 · Answer 9 · 2020-04-28 05:56:01

68

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How to prove $\lim_{n\to\infty}\frac{1}{\Gamma(n+1/2)}\int_{n}^{\infty} (\sqrt{t}-\sqrt{n}) \exp(-t)t^{n-1} dt = 0$?

Published on 28 Apr 2020 - 5:56

score 431 · Answer 10 · 2020-04-28 14:10:13

431

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How to prove $\lim_{n\to\infty}\frac{1}{\Gamma(n)}\int_{n}^{\infty}t^{n-1}e^{-t}dt = 1/2$?

Published on 28 Apr 2020 - 14:10

score 2.1k · Answer 11 · 2020-04-28 19:46:15

2.1k

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Fourier series of a polynomial.

Published on 28 Apr 2020 - 19:46

score 61 · Answer 12 · 2026-03-30 02:07:47

61

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Unexplained points of intersection of $f(x)$ and $f(\sin(\pi x)+x)$

Published on 30 Mar 2026 - 2:07

score 190 · Answer 13 · 2026-03-25 15:53:39

190

Views

Solving $f(x+1)=xf(x)f\left(\frac1x\right)$

Published on 25 Mar 2026 - 15:53

score 126 · Answer 14 · 2026-03-25 22:26:42

126

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Is there a way to simplify the solution to $\int_{1}^{e^{\frac{1}{e}}} x^{x^{x^{x^{...}}}} dx$

Published on 25 Mar 2026 - 22:26

score 97 · Answer 15 · 2026-03-25 12:19:22

97

Views

How can we prove that $\int_0^\infty e^{-ix}x^{s-1} \ \mathrm{d}x = i^{-s}\Gamma(s)?$

Published on 25 Mar 2026 - 12:19

List Question

How to evaluate $\int_0^\infty \frac{x^5(e^{3x}-e^x)}{(e^x-1)^4}\,\mathrm{d}x$

Value of $\sum_{n=1}^{\infty}\frac{1}{\sqrt{n!}}$?

Asymptotic Behaviour of $\Gamma^{(k)}(1)$

Branch of incomplete gamma function

What material do I need to cover to understand the gamma function?

Gamma Function Specifics and properties

How to prove $\lim_{n\to\infty}\frac{1}{\Gamma(n/2+1)}\int_{0}^{n} t^{n/2}e^{-t}dt = 1$?

Show $ \int_0^\infty \ x^{\alpha-1}e^{-x/\beta} \ dx = \beta^\alpha\Gamma(\alpha).$

How to prove $\lim_{n\to\infty}\frac{1}{\Gamma(n+1/2)}\int_{n}^{\infty} (\sqrt{t}-\sqrt{n}) \exp(-t)t^{n-1} dt = 0$?

How to prove $\lim_{n\to\infty}\frac{1}{\Gamma(n)}\int_{n}^{\infty}t^{n-1}e^{-t}dt = 1/2$?

Fourier series of a polynomial.

Unexplained points of intersection of $f(x)$ and $f(\sin(\pi x)+x)$

Solving $f(x+1)=xf(x)f\left(\frac1x\right)$

Is there a way to simplify the solution to $\int_{1}^{e^{\frac{1}{e}}} x^{x^{x^{x^{...}}}} dx$

How can we prove that $\int_0^\infty e^{-ix}x^{s-1} \ \mathrm{d}x = i^{-s}\Gamma(s)?$

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