Question

How to prove that $\int_{0}^{\infty}{\sin^4(x)\ln(x)}\cdot{\mathrm dx\over x^2}={\pi\over 4}\cdot(1-\gamma)?$

score 272 · Answer 1 · 2026-04-16 05:16:03

272

Views

How to prove that $\int_{0}^{\infty}{\sin^4(x)\ln(x)}\cdot{\mathrm dx\over x^2}={\pi\over 4}\cdot(1-\gamma)?$

Published on 16 Apr 2026 - 5:16

score 109 · Answer 2 · 2026-04-09 10:15:50

109

Views

How to evaluate this logarithmic integral?

Published on 09 Apr 2026 - 10:15

score 37 · Answer 3 · 2026-04-14 11:42:46

37

Views

Does $\int_1^\infty x^{-x}$ converges?

Published on 14 Apr 2026 - 11:42

score 231 · Answer 4 · 2026-04-16 12:51:13

231

Views

How to prove $\int_{-1}^{1}{\mathrm dx\over x}\cdot\sqrt{1+x\over 1-x}\cdot{2x(x-1)+\ln((1-x)(1+x)^3)\over x}=\pi^2-4\pi?$

Published on 16 Apr 2026 - 12:51

score 38 · Answer 5 · 2026-04-14 14:50:55

38

Views

Lacking understanding of some properties of the improper integral

Published on 14 Apr 2026 - 14:50

score 20 · Answer 6 · 2026-04-14 05:39:29

20

Views

Generalization of improper integrals under a closed interval to the regular integral

Published on 14 Apr 2026 - 5:39

score 123 · Answer 7 · 2026-04-09 01:43:23

123

Views

Improper integral over an open interval using the comparison test

Published on 09 Apr 2026 - 1:43

score 125 · Answer 8 · 2026-04-16 21:57:12

125

Views

Convergence of $\int^\infty_0\frac{1}{1+x^2\sin^2(5x)}\,dx$

Published on 16 Apr 2026 - 21:57

score 158 · Answer 9 · 2026-04-17 09:03:51

158

Views

Does $\int_1^2\frac{\sin{x}}{\log{x}} \, dx$ converge?

Published on 17 Apr 2026 - 9:03

score 2.5k · Answer 10 · 2026-04-14 16:21:24

2.5k

Views

For what values of p does the improper integral converge?

Published on 14 Apr 2026 - 16:21

score 180 · Answer 11 · 2026-04-16 19:27:46

180

Views

Why is it that $\int_{-\infty}^{\infty} \frac{x}{1+x^2} dx $ diverges, while $\int_{-\infty}^{\infty} \frac{x}{1+x^8} dx $ converges.

Published on 16 Apr 2026 - 19:27

score 789 · Answer 12 · 2026-04-10 02:50:17

789

Views

An almost Fresnel integral

Published on 10 Apr 2026 - 2:50

score 198 · Answer 13 · 2026-04-14 13:47:57

198

Views

Is there a way to prove convergence of an improper integral without calculating the integral itself?

Published on 14 Apr 2026 - 13:47

score 193 · Answer 14 · 2026-04-16 13:33:54

193

Views

Determine $\int^{\infty}_1 \frac1{x^2+x} \mathrm d x$

Published on 16 Apr 2026 - 13:33

score 111 · Answer 15 · 2026-04-13 14:40:34

111

Views

Integral $ \int_a^\infty \frac{dr}{r^5 \sqrt{r-a}}$

Published on 13 Apr 2026 - 14:40

List Question

How to prove that $\int_{0}^{\infty}{\sin^4(x)\ln(x)}\cdot{\mathrm dx\over x^2}={\pi\over 4}\cdot(1-\gamma)?$

How to evaluate this logarithmic integral?

Does $\int_1^\infty x^{-x}$ converges?

How to prove $\int_{-1}^{1}{\mathrm dx\over x}\cdot\sqrt{1+x\over 1-x}\cdot{2x(x-1)+\ln((1-x)(1+x)^3)\over x}=\pi^2-4\pi?$

Lacking understanding of some properties of the improper integral

Generalization of improper integrals under a closed interval to the regular integral

Improper integral over an open interval using the comparison test

Convergence of $\int^\infty_0\frac{1}{1+x^2\sin^2(5x)}\,dx$

Does $\int_1^2\frac{\sin{x}}{\log{x}} \, dx$ converge?

For what values of p does the improper integral converge?

Why is it that $\int_{-\infty}^{\infty} \frac{x}{1+x^2} dx $ diverges, while $\int_{-\infty}^{\infty} \frac{x}{1+x^8} dx $ converges.

An almost Fresnel integral

Is there a way to prove convergence of an improper integral without calculating the integral itself?

Determine $\int^{\infty}_1 \frac1{x^2+x} \mathrm d x$

Integral $ \int_a^\infty \frac{dr}{r^5 \sqrt{r-a}}$

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