Question

Given $\int_0^{\infty}e^{-x^2}dx = \frac{\sqrt{\pi}}{2}$, evaluate $\int_0^{\infty}e^{-a^2x^2-\frac{b^2}{x^2}}dx $

score 281 · Answer 1 · 2026-04-15 22:52:02

281

Views

Given $\int_0^{\infty}e^{-x^2}dx = \frac{\sqrt{\pi}}{2}$, evaluate $\int_0^{\infty}e^{-a^2x^2-\frac{b^2}{x^2}}dx $

Published on 15 Apr 2026 - 22:52

score 219 · Answer 2 · 2026-04-12 06:52:45

219

Views

Integrals of a function and its absolute value

Published on 12 Apr 2026 - 6:52

score 451 · Answer 3 · 2026-04-17 10:16:41

451

Views

Prove $\int_0^\infty \exp (-x^2) dx=\int_0^\infty \exp \left(-x^2 \left( 1-\frac{4}{x^2-2}\right)^2 \right) dx$

Published on 17 Apr 2026 - 10:16

score 430 · Answer 4 · 2026-04-17 11:53:44

430

Views

Prove $\int_0^{\infty} \frac{x^2}{\cosh^2 (x^2)} dx=\frac{\sqrt{2}-2}{4} \sqrt{\pi}~ \zeta \left( \frac{1}{2} \right)$

Published on 17 Apr 2026 - 11:53

score 900 · Answer 5 · 2026-04-15 13:39:28

900

Views

Find the values of $a$ such that the integral converges.

Published on 15 Apr 2026 - 13:39

score 177 · Answer 6 · 2026-04-16 02:35:47

177

Views

Prove $\int_0^{\infty} \int_0^{\infty} \frac{\sqrt{xy} ~dxdy}{(x+y)(1+x y)^s}=\frac{\pi}{2(s-1)}$ for $s>1$

Published on 16 Apr 2026 - 2:35

score 1k · Answer 7 · 2026-04-12 00:37:07

1k

Views

A novelty integral for $\pi$

Published on 12 Apr 2026 - 0:37

score 489 · Answer 8 · 2026-04-13 04:06:50

489

Views

integral inequality for $f(x)$ and $f(\sqrt{x})$

Published on 13 Apr 2026 - 4:06

score 137 · Answer 9 · 2026-04-11 19:16:17

137

Views

Integral test application

Published on 11 Apr 2026 - 19:16

score 283 · Answer 10 · 2026-04-16 14:33:50

283

Views

Prove that $\int_{0}^{1}{(1-x)(x-3)\over 1+x^2}\cdot{dx\over \ln{x}}={\ln{8\over \Gamma^4(3/4)}}$

Published on 16 Apr 2026 - 14:33

score 128 · Answer 11 · 2026-04-16 14:02:54

128

Views

An upper bound of $ \left| \frac{1}{s}\log\zeta(s) \right| $ for $\Re s=\sigma>1$, from this integral formula and a related comparison

Published on 16 Apr 2026 - 14:02

score 425 · Answer 12 · 2026-04-16 07:15:10

425

Views

Evaluate the integral $ \int_0^{+\infty} \frac{\sin(x^2)}{x^4+1} dx $ using the residue method

Published on 16 Apr 2026 - 7:15

score 474 · Answer 13 · 2026-04-17 00:29:23

474

Views

Determine whether the following integral is convergent or divergent $\int^{2}_{0} \frac{1}{\sqrt[3]{x} (x+\sqrt{x})}dx$

Published on 17 Apr 2026 - 0:29

score 104 · Answer 14 · 2026-04-16 10:51:27

104

Views

$\int_0^\infty \frac{ x^{1/3}}{(x+a)(x+b)} dx $

Published on 16 Apr 2026 - 10:51

score 68 · Answer 15 · 2026-04-09 10:40:53

68

Views

Confusion about the convergence of Riemann zeta function in terms of the integral

Published on 09 Apr 2026 - 10:40

List Question

Given $\int_0^{\infty}e^{-x^2}dx = \frac{\sqrt{\pi}}{2}$, evaluate $\int_0^{\infty}e^{-a^2x^2-\frac{b^2}{x^2}}dx $

Integrals of a function and its absolute value

Prove $\int_0^\infty \exp (-x^2) dx=\int_0^\infty \exp \left(-x^2 \left( 1-\frac{4}{x^2-2}\right)^2 \right) dx$

Prove $\int_0^{\infty} \frac{x^2}{\cosh^2 (x^2)} dx=\frac{\sqrt{2}-2}{4} \sqrt{\pi}~ \zeta \left( \frac{1}{2} \right)$

Find the values of $a$ such that the integral converges.

Prove $\int_0^{\infty} \int_0^{\infty} \frac{\sqrt{xy} ~dxdy}{(x+y)(1+x y)^s}=\frac{\pi}{2(s-1)}$ for $s>1$

A novelty integral for $\pi$

integral inequality for $f(x)$ and $f(\sqrt{x})$

Integral test application

Prove that $\int_{0}^{1}{(1-x)(x-3)\over 1+x^2}\cdot{dx\over \ln{x}}={\ln{8\over \Gamma^4(3/4)}}$

An upper bound of $ \left| \frac{1}{s}\log\zeta(s) \right| $ for $\Re s=\sigma>1$, from this integral formula and a related comparison

Evaluate the integral $ \int_0^{+\infty} \frac{\sin(x^2)}{x^4+1} dx $ using the residue method

Determine whether the following integral is convergent or divergent $\int^{2}_{0} \frac{1}{\sqrt[3]{x} (x+\sqrt{x})}dx$

$\int_0^\infty \frac{ x^{1/3}}{(x+a)(x+b)} dx $

Confusion about the convergence of Riemann zeta function in terms of the integral

Trending Questions

Popular # Hahtags

Popular Questions