Question

Proving $\int_0^\infty x^ne^{-tx}\frac{\sin x}xdx=\frac{\sin n\theta}{(1+t^2)^{n/2}}(n-1)!$, where $\theta=\arcsin\frac1{\sqrt{1+t^2}}$

score 137 · Answer 1 · 2026-03-27 23:35:24

137

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Proving $\int_0^\infty x^ne^{-tx}\frac{\sin x}xdx=\frac{\sin n\theta}{(1+t^2)^{n/2}}(n-1)!$, where $\theta=\arcsin\frac1{\sqrt{1+t^2}}$

Published on 27 Mar 2026 - 23:35

score 66 · Answer 2 · 2019-02-13 15:08:48

66

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For which $p,q$ does the $\int_0^{\infty} \frac{x^p}{\mid{1-x}\mid^q}dx$ exist?

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662

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58

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63

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123

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134

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score 80 · Answer 11 · 2019-02-16 17:37:35

80

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Classify $I(a,b)=\int_{0}^{\frac{\pi}{2}} \frac{dx}{\sin(x)^a\cos(x)^b}$ with $a,b\in \mathbb{R}$

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score 140 · Answer 12 · 2019-02-17 10:12:52

140

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score 118 · Answer 13 · 2026-03-29 13:21:40

118

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score 81 · Answer 14 · 2019-02-18 19:29:55

81

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Published on 18 Feb 2019 - 19:29

score 186 · Answer 15 · 2019-02-19 14:21:41

186

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List Question

Proving $\int_0^\infty x^ne^{-tx}\frac{\sin x}xdx=\frac{\sin n\theta}{(1+t^2)^{n/2}}(n-1)!$, where $\theta=\arcsin\frac1{\sqrt{1+t^2}}$

For which $p,q$ does the $\int_0^{\infty} \frac{x^p}{\mid{1-x}\mid^q}dx$ exist?

Ramanujan's Master Theorem relation to Analytic Continuation

A naive change of integration limit

Improper Integrals in Analysis

Problem in solving the improper integral.

Non-uniform convergence of improper integrals II

Integral $\int_{0}^{1}\frac{\log x}{2 - x} dx$

Studying convergence of an integral at the vary of alpha

Asymptotic behavior of the integral $H(\beta)=\frac{2}{\pi}\beta\int_{0}^{\infty}\exp\left(-x^{3/2}\right)\sin(\beta x)xdx$

Classify $I(a,b)=\int_{0}^{\frac{\pi}{2}} \frac{dx}{\sin(x)^a\cos(x)^b}$ with $a,b\in \mathbb{R}$

Integral $\int\limits_{0}^{\infty}e^{-\frac{1}{2}\left(y^2+\frac{t^2}{y^2}\right)}\,dy$

Bounded from above integral with exponential

Show $\int \frac{\sin(x^p)}{x} dx = \frac{\operatorname{Si}(x)}{p} $

Triple integral convergence

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