Question

Showing $\int_{-\infty}^{\infty} \exp\left({-a x^2 -\frac{b}{x^2}}\right) \,dx = \sqrt{\frac{\pi}{a}}e^{-2\sqrt{ab} }$

score 112 · Answer 1 · 2026-04-14 11:34:15

112

Views

Showing $\int_{-\infty}^{\infty} \exp\left({-a x^2 -\frac{b}{x^2}}\right) \,dx = \sqrt{\frac{\pi}{a}}e^{-2\sqrt{ab} }$

Published on 14 Apr 2026 - 11:34

score 246 · Answer 2 · 2026-04-15 09:28:24

246

Views

For which $\alpha$ does the integral $\int\limits_2^{+\infty}\frac{e^{\alpha x}}{(x-1)^{\alpha}\ln x}\mathrm{d}x$ converge?

Published on 15 Apr 2026 - 9:28

score 61 · Answer 3 · 2026-04-16 01:24:54

61

Views

Calculate the integral or prove that it diverges

Published on 16 Apr 2026 - 1:24

score 105 · Answer 4 · 2026-04-10 10:09:29

105

Views

Recurring Exponential Integral

Published on 10 Apr 2026 - 10:09

score 565 · Answer 5 · 2026-05-04 13:44:28

565

Views

Fourier series representation of $\log^2\cos(x)$

Published on 04 May 2026 - 13:44

score 228 · Answer 6 · 2026-04-17 08:08:23

228

Views

evaluation of a double integral over the first quadrant

Published on 17 Apr 2026 - 8:08

score 131 · Answer 7 · 2026-03-27 20:54:01

131

Views

Evaluating an integral involving modified Bessel functions $K_{i\Omega}$

Published on 27 Mar 2026 - 20:54

score 292 · Answer 8 · 2026-04-22 07:16:29

292

Views

How to evaluate $\int_{-\infty}^{\infty}\frac{\sin^3(x)}{x^3}\mathrm{d}x$ via the complex epsilon method?

Published on 22 Apr 2026 - 7:16

score 1.3k · Answer 9 · 2026-04-11 09:42:12

1.3k

Views

Limit of uniformly continuous functions with convergent improper integrals.

Published on 11 Apr 2026 - 9:42

score 125 · Answer 10 · 2026-04-16 13:16:01

125

Views

Prove that $\int_0^\infty xf(x) dx = a$ and $0<f<1$ implies $\int_0^\infty xf(x) dx > \frac{a^2}{2}$

Published on 16 Apr 2026 - 13:16

score 130 · Answer 11 · 2026-04-16 05:27:51

130

Views

About the integral $\int_\mathbb{R} e^{-x^2 -(x-\xi)^2}\,dx$

Published on 16 Apr 2026 - 5:27

score 59 · Answer 12 · 2026-04-16 11:42:42

59

Views

Solve a definite integral with a gaussian integrand.

Published on 16 Apr 2026 - 11:42

score 63 · Answer 13 · 2026-04-09 01:27:47

63

Views

How do I justify the integral over a limit

Published on 09 Apr 2026 - 1:27

score 327 · Answer 14 · 2026-04-13 07:05:39

327

Views

Integral over three Bessel functions $J_{0} ( \alpha\ x ) K_{iP} ( \beta x ) K_{iP} ( \xi x )$

Published on 13 Apr 2026 - 7:05

score 74 · Answer 15 · 2026-04-14 15:49:02

74

Views

Why does this converge or diverge? (Improper Integral)

Published on 14 Apr 2026 - 15:49

List Question

Showing $\int_{-\infty}^{\infty} \exp\left({-a x^2 -\frac{b}{x^2}}\right) \,dx = \sqrt{\frac{\pi}{a}}e^{-2\sqrt{ab} }$

For which $\alpha$ does the integral $\int\limits_2^{+\infty}\frac{e^{\alpha x}}{(x-1)^{\alpha}\ln x}\mathrm{d}x$ converge?

Calculate the integral or prove that it diverges

Recurring Exponential Integral

Fourier series representation of $\log^2\cos(x)$

evaluation of a double integral over the first quadrant

Evaluating an integral involving modified Bessel functions $K_{i\Omega}$

How to evaluate $\int_{-\infty}^{\infty}\frac{\sin^3(x)}{x^3}\mathrm{d}x$ via the complex epsilon method?

Limit of uniformly continuous functions with convergent improper integrals.

Prove that $\int_0^\infty xf(x) dx = a$ and $0<f<1$ implies $\int_0^\infty xf(x) dx > \frac{a^2}{2}$

About the integral $\int_\mathbb{R} e^{-x^2 -(x-\xi)^2}\,dx$

Solve a definite integral with a gaussian integrand.

How do I justify the integral over a limit

Integral over three Bessel functions $J_{0} ( \alpha\ x ) K_{iP} ( \beta x ) K_{iP} ( \xi x )$

Why does this converge or diverge? (Improper Integral)

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