Question

Definite integral including natural log, cosine, and hyperbolic sine

score 95 · Answer 1 · 2026-04-16 05:58:47

95

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Definite integral including natural log, cosine, and hyperbolic sine

Published on 16 Apr 2026 - 5:58

score 1.7k · Answer 2 · 2026-04-15 22:11:09

1.7k

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How to compute $\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$?

Published on 15 Apr 2026 - 22:11

score 149 · Answer 3 · 2026-04-17 00:59:15

149

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How to compute $\int_0^1 \frac{x-1}{\ln(x)} dx = \ln(2)$? and $\int_0^\infty \ln(t) e^{-t} dt $?

Published on 17 Apr 2026 - 0:59

score 185 · Answer 4 · 2026-04-16 04:15:06

185

Views

How to prove $\int_0^1 \ln\left(\frac{1+x}{1-x}\right) \frac{dx}{x} = \frac{\pi^2}{4}$?

Published on 16 Apr 2026 - 4:15

score 1k · Answer 5 · 2026-04-11 06:59:50

1k

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Convergence of a double integral

Published on 11 Apr 2026 - 6:59

score 116 · Answer 6 · 2026-04-17 14:13:08

116

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$\int _{ 0 }^{ 1 }{ \frac { { x }^{ t }-1 }{ \ln { x } } dx } $

Published on 17 Apr 2026 - 14:13

score 178 · Answer 7 · 2026-04-15 17:10:31

178

Views

$ \int_{-\infty }^{\infty} e^{-x^{2} + sin x}dx$

Published on 15 Apr 2026 - 17:10

score 87 · Answer 8 · 2026-04-16 15:20:15

87

Views

Given $|f(x)|=1$,how to construct an $f(x)$, such that $\int ^{+\infty }_{0}f\left( x\right) dx$ converges

Published on 16 Apr 2026 - 15:20

score 272 · Answer 9 · 2026-04-16 08:29:58

272

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Find $f$ so that $\int_{1}^{\infty}f(x)dx$ exists, but $\displaystyle \lim_{x\to\infty}f(x)$ does not exist?

Published on 16 Apr 2026 - 8:29

score 66 · Answer 10 · 2026-04-17 13:59:46

66

Views

For what $\alpha$ does the integral absolutely and for what conditionally converge?

Published on 17 Apr 2026 - 13:59

score 124 · Answer 11 · 2026-04-16 11:53:43

124

Views

Find functions $f$ and $\alpha$ such that $\int_1^{\infty}|f|d\alpha$ converges, but $\int_1^{\infty}fd\alpha$ does not exist?

Published on 16 Apr 2026 - 11:53

score 78 · Answer 12 · 2026-04-14 06:20:42

78

Views

A real integral (may be requires contour integration)?

Published on 14 Apr 2026 - 6:20

score 39 · Answer 13 · 2026-04-14 12:52:28

39

Views

Parametric integral and equivalence in $\infty$

Published on 14 Apr 2026 - 12:52

score 143 · Answer 14 · 2026-04-13 22:13:27

143

Views

Improper Integral: $\int_{-\infty}^\infty\frac{e^{-t}}{1+e^{-2t}}\ dt$

Published on 13 Apr 2026 - 22:13

score 349 · Answer 15 · 2026-04-09 05:24:52

349

Views

Decomposition into partial fractions to compute an integral

Published on 09 Apr 2026 - 5:24

List Question

Definite integral including natural log, cosine, and hyperbolic sine

How to compute $\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$?

How to compute $\int_0^1 \frac{x-1}{\ln(x)} dx = \ln(2)$? and $\int_0^\infty \ln(t) e^{-t} dt $?

How to prove $\int_0^1 \ln\left(\frac{1+x}{1-x}\right) \frac{dx}{x} = \frac{\pi^2}{4}$?

Convergence of a double integral

$\int _{ 0 }^{ 1 }{ \frac { { x }^{ t }-1 }{ \ln { x } } dx } $

$ \int_{-\infty }^{\infty} e^{-x^{2} + sin x}dx$

Given $|f(x)|=1$,how to construct an $f(x)$, such that $\int ^{+\infty }_{0}f\left( x\right) dx$ converges

Find $f$ so that $\int_{1}^{\infty}f(x)dx$ exists, but $\displaystyle \lim_{x\to\infty}f(x)$ does not exist?

For what $\alpha$ does the integral absolutely and for what conditionally converge?

Find functions $f$ and $\alpha$ such that $\int_1^{\infty}|f|d\alpha$ converges, but $\int_1^{\infty}fd\alpha$ does not exist?

A real integral (may be requires contour integration)?

Parametric integral and equivalence in $\infty$

Improper Integral: $\int_{-\infty}^\infty\frac{e^{-t}}{1+e^{-2t}}\ dt$

Decomposition into partial fractions to compute an integral

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