Question

How to prove that $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists

score 270 · Answer 1 · 2014-08-21 01:54:46

270

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How to prove that $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists

Published on 21 Aug 2014 - 1:54

score 2k · Answer 2 · 2026-03-26 19:02:05

2k

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Intuitive reason for why the Gaussian integral converges to the square root of pi?

Published on 26 Mar 2026 - 19:02

score 566 · Answer 3 · 2014-08-24 18:34:24

566

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cosine integral

Published on 24 Aug 2014 - 18:34

score 39.1k · Answer 4 · 2026-03-26 12:48:30

39.1k

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How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$

Published on 26 Mar 2026 - 12:48

score 141 · Answer 5 · 2014-08-25 04:05:58

141

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For which values is this improper integral convergent?

Published on 25 Aug 2014 - 4:05

score 73 · Answer 6 · 2014-08-25 10:34:02

73

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Convergence of $\int_0^{\infty}\left(\frac{x}{\ln x}\right)^c\frac{1}{1+x^{4c}}\ dx$

Published on 25 Aug 2014 - 10:34

score 30 · Answer 7 · 2014-08-26 10:34:33

30

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Comparison of two integrals in $\Bbb R$

Published on 26 Aug 2014 - 10:34

score 1k · Answer 8 · 2026-03-26 06:07:01

1k

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Integral of $\ln(x)\operatorname{sech}(x)$

Published on 26 Mar 2026 - 6:07

score 128 · Answer 9 · 2014-08-27 05:12:20

128

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Value of convergence of $\displaystyle\int\frac{\sqrt{x}}{1+x^2}$

Published on 27 Aug 2014 - 5:12

score 264 · Answer 10 · 2026-04-03 05:21:04

264

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How to take the limit of the improper integral of a sequence of functions

Published on 03 Apr 2026 - 5:21

score 427 · Answer 11 · 2014-08-30 19:38:31

427

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A closed-form of $\frac{1}{2}\int_0^\infty\left[\frac{x^2\cos x}{\cosh 2x-\cos x}-\frac{2x^2}{e^{4x}-2e^{2x}\cos x+1}\right]\,dx$

Published on 30 Aug 2014 - 19:38

score 641 · Answer 12 · 2014-08-31 17:37:36

641

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Convergence of Integral near 0

Published on 31 Aug 2014 - 17:37

score 2.7k · Answer 13 · 2026-03-29 03:28:27

2.7k

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Prove $\int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2 $

Published on 29 Mar 2026 - 3:28

score 1.1k · Answer 14 · 2014-09-01 16:27:34

1.1k

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Prove that $\int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}$

Published on 01 Sep 2014 - 16:27

score 456 · Answer 15 · 2026-04-03 15:39:45

456

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$\int_{0}^{\infty} \frac{\cos(x)}{1+x^2} dx$ and $\int_{0}^{\infty} \frac {\ln(x)}{x^2+b^2} dx$

Published on 03 Apr 2026 - 15:39

List Question

How to prove that $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists

Intuitive reason for why the Gaussian integral converges to the square root of pi?

cosine integral

How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$

For which values is this improper integral convergent?

Convergence of $\int_0^{\infty}\left(\frac{x}{\ln x}\right)^c\frac{1}{1+x^{4c}}\ dx$

Comparison of two integrals in $\Bbb R$

Integral of $\ln(x)\operatorname{sech}(x)$

Value of convergence of $\displaystyle\int\frac{\sqrt{x}}{1+x^2}$

How to take the limit of the improper integral of a sequence of functions

A closed-form of $\frac{1}{2}\int_0^\infty\left[\frac{x^2\cos x}{\cosh 2x-\cos x}-\frac{2x^2}{e^{4x}-2e^{2x}\cos x+1}\right]\,dx$

Convergence of Integral near 0

Prove $\int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2 $

Prove that $\int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}$

$\int_{0}^{\infty} \frac{\cos(x)}{1+x^2} dx$ and $\int_{0}^{\infty} \frac {\ln(x)}{x^2+b^2} dx$

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