Question

Conditional/Absolute convergence of $\int_{1}^{\infty}\cos(x^{2})\,\mathrm dx$

score 665 · Answer 1 · 2026-04-16 03:50:13

665

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Conditional/Absolute convergence of $\int_{1}^{\infty}\cos(x^{2})\,\mathrm dx$

Published on 16 Apr 2026 - 3:50

score 142 · Answer 2 · 2026-04-17 01:25:17

142

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Proving $\int\limits _0^\infty\frac {\sin^{2n+1}x}{x}dx=\frac 1 {4^n}\binom{2n}{n}\int_0^\infty \frac {\sin{x}}{x}dx$

Published on 17 Apr 2026 - 1:25

score 182 · Answer 3 · 2026-04-22 21:55:12

182

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Prove that if $f_n\to f$ uniformly and $\int_0^\infty|f_n|\le M$, then $\int_0^\infty|f|<\infty$

Published on 22 Apr 2026 - 21:55

score 412 · Answer 4 · 2026-04-16 04:53:08

412

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Convergence of $\int\limits_{-\infty}^{\infty}\frac{\sin{x^2}}{x}dx$

Published on 16 Apr 2026 - 4:53

score 102 · Answer 5 · 2026-04-17 14:38:32

102

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Evaluating $\int_0^{\infty} \log\left(1-2\frac{\cos(2a)}{x^2}+\frac{1}{x^4}\right)^2 \ dx, \quad 0\le a \le \pi $

Published on 17 Apr 2026 - 14:38

score 169 · Answer 6 · 2026-04-16 01:39:41

169

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Why does $\int\limits_0^1 {\dfrac{{x - 1}}{{\ln x}}} \;\text{d}x=\ln2$?

Published on 16 Apr 2026 - 1:39

score 2k · Answer 7 · 2026-04-16 14:51:19

2k

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convergence test : $\int_{0}^\infty \mathrm 1/(x\ln(x)^2)\,\mathrm dx $

Published on 16 Apr 2026 - 14:51

score 345 · Answer 8 · 2026-04-17 09:49:54

345

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Evaluate $\int \limits_{0}^{\infty} \frac{x}{1+x^2} dx$

Published on 17 Apr 2026 - 9:49

score 4.7k · Answer 9 · 2026-04-15 12:46:01

4.7k

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How do I show that $\int_0^\infty \frac{\sin(ax) \sin(bx)}{x^{2}} \, \mathrm dx = \pi \min(a,b)/2$

Published on 15 Apr 2026 - 12:46

score 389 · Answer 10 · 2026-04-17 14:00:28

389

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Frullani version of the classic $\log \left( 1 + 2\alpha \cos px + \alpha^2\right)$ integral.

Published on 17 Apr 2026 - 14:00

score 4.1k · Answer 11 · 2026-04-12 04:01:22

4.1k

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Dirichlet integral

Published on 12 Apr 2026 - 4:01

score 191 · Answer 12 · 2026-04-23 02:39:04

191

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$F(u)= \frac{2}{\pi}\int_{0}^\infty \frac{uf(x)}{u^2 + x^2}dx.$ Show that $\lim\limits_{u\downarrow0}F(u)=f(0)$.

Published on 23 Apr 2026 - 2:39

score 396 · Answer 13 · 2026-04-11 12:40:46

396

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Improper integral depending of one parameter

Published on 11 Apr 2026 - 12:40

score 664 · Answer 14 · 2026-04-14 16:04:12

664

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A 'complicated' integral: $ \int \limits_{-\infty}^{\infty}\frac{\sin(x)}{x}$

Published on 14 Apr 2026 - 16:04

score 5.1k · Answer 15 · 2026-04-12 17:27:32

5.1k

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A method for calculating this integral hermite polynomials

Published on 12 Apr 2026 - 17:27

List Question

Conditional/Absolute convergence of $\int_{1}^{\infty}\cos(x^{2})\,\mathrm dx$

Proving $\int\limits _0^\infty\frac {\sin^{2n+1}x}{x}dx=\frac 1 {4^n}\binom{2n}{n}\int_0^\infty \frac {\sin{x}}{x}dx$

Prove that if $f_n\to f$ uniformly and $\int_0^\infty|f_n|\le M$, then $\int_0^\infty|f|<\infty$

Convergence of $\int\limits_{-\infty}^{\infty}\frac{\sin{x^2}}{x}dx$

Evaluating $\int_0^{\infty} \log\left(1-2\frac{\cos(2a)}{x^2}+\frac{1}{x^4}\right)^2 \ dx, \quad 0\le a \le \pi $

Why does $\int\limits_0^1 {\dfrac{{x - 1}}{{\ln x}}} \;\text{d}x=\ln2$?

convergence test : $\int_{0}^\infty \mathrm 1/(x\ln(x)^2)\,\mathrm dx $

Evaluate $\int \limits_{0}^{\infty} \frac{x}{1+x^2} dx$

How do I show that $\int_0^\infty \frac{\sin(ax) \sin(bx)}{x^{2}} \, \mathrm dx = \pi \min(a,b)/2$

Frullani version of the classic $\log \left( 1 + 2\alpha \cos px + \alpha^2\right)$ integral.

Dirichlet integral

$F(u)= \frac{2}{\pi}\int_{0}^\infty \frac{uf(x)}{u^2 + x^2}dx.$ Show that $\lim\limits_{u\downarrow0}F(u)=f(0)$.

Improper integral depending of one parameter

A 'complicated' integral: $ \int \limits_{-\infty}^{\infty}\frac{\sin(x)}{x}$

A method for calculating this integral hermite polynomials

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