Question

Show $\int_{0}^{\infty}\frac{\sin x^{2}}{\pi-x^{2}}dx=\frac{\sqrt{\pi}}{2}\left[C(\sqrt{2})+S(\sqrt{2})\right]$

score 170 · Answer 1 · 2024-02-03 06:44:08

170

Views

Show $\int_{0}^{\infty}\frac{\sin x^{2}}{\pi-x^{2}}dx=\frac{\sqrt{\pi}}{2}\left[C(\sqrt{2})+S(\sqrt{2})\right]$

Published on 03 Feb 2024 - 6:44

score 76 · Answer 2 · 2024-02-05 09:18:40

76

Views

Probability that two random integers have at most $n$ prime factors in common

Published on 05 Feb 2024 - 9:18

score 24 · Answer 3 · 2026-03-28 06:48:25

24

Views

Calculate probability of Gaussian random variable exceeding threshold when sampled from different Gaussian distribution

Published on 28 Mar 2026 - 6:48

score 392 · Answer 4 · 2024-02-07 07:13:14

392

Views

Integration of $ \int_{0}^{\frac{\pi}{2}} x \log(1-\cos x) \,dx $

Published on 07 Feb 2024 - 7:13

score 35 · Answer 5 · 2026-04-03 00:22:13

35

Views

"Nice" form/bounds for $\ln\prod_{i=1}^n \left(1 + \frac{x_i^2}{n}\right)$

Published on 03 Apr 2026 - 0:22

score 205 · Answer 6 · 2024-02-10 07:38:16

205

Views

How to calculate this integral $\int_{0}^{\infty} \left(\frac{x - 1}{\ln(x)}\right)^2 \cdot \frac{1}{1 + x^n} \,dx$

Published on 10 Feb 2024 - 7:38

score 31 · Answer 7 · 2026-03-25 21:22:16

31

Views

Is there a Closed-Form Solution for L2 Regularization Raised to a Power?

Published on 25 Mar 2026 - 21:22

score 89 · Answer 8 · 2024-02-11 01:05:06

89

Views

Integrating $\int_{0}^{\infty} \frac{\tanh \left(\frac{x}{2}\right)+\tanh (2 x)}{x}\left(e^{\frac{3 x}{2}}-1\right)^{2} e^{-4 x} \mathrm{~d}x$

Published on 11 Feb 2024 - 1:05

score 216 · Answer 9 · 2026-03-25 06:07:08

216

Views

How to calculate this sum $\sum_{n=1}^{\infty} \frac{H_n \cdot H_{n+1}}{(n+1)(n+2)}$

Published on 25 Mar 2026 - 6:07

score 151 · Answer 10 · 2024-02-11 18:54:08

151

Views

Integrating $\int_{0}^{1} \left(\frac{\arctan(x) - x}{x^2}\right)^2 \,dx$

Published on 11 Feb 2024 - 18:54

score 107 · Answer 11 · 2024-02-11 20:32:57

107

Views

How to integrate $\int_{0}^{\infty} \frac{1}{x^2 (\tan^2 x + \cot^2 x)} \,dx$

Published on 11 Feb 2024 - 20:32

score 179 · Answer 12 · 2026-04-02 17:43:41

179

Views

Closed form for $A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$

Published on 02 Apr 2026 - 17:43

score 210 · Answer 13 · 2024-02-12 07:12:17

210

Views

How to integrate $\int_{0}^{1} \int_{0}^{1} \int_{0}^{1}\frac{x^{4a - 1} \ln(x)}{\sqrt{yz} \cdot (1 + x^{2a}z + yzx^{4a} + yx^{2a})} \,dx \,dy\,dz$

Published on 12 Feb 2024 - 7:12

score 163 · Answer 14 · 2024-02-12 11:41:38

163

Views

How to integrate $\int_{0}^{1} \int_{0}^{1} \ln\left(\frac{1}{\sinh^2(x) + \cosh^2(y)}\right) \,dx\,dy$

Published on 12 Feb 2024 - 11:41

score 70 · Answer 15 · 2024-02-13 06:10:40

70

Views

How to integrate $\int_{0}^{1} \int_{0}^{\infty} \frac{\ln(1 + x^3) \cdot (\ln(1 + y) + \arctan(y))}{(1 + x^2)(1 + y)^2} \,dx\,dy$

Published on 13 Feb 2024 - 6:10

List Question

Show $\int_{0}^{\infty}\frac{\sin x^{2}}{\pi-x^{2}}dx=\frac{\sqrt{\pi}}{2}\left[C(\sqrt{2})+S(\sqrt{2})\right]$

Probability that two random integers have at most $n$ prime factors in common

Calculate probability of Gaussian random variable exceeding threshold when sampled from different Gaussian distribution

Integration of $ \int_{0}^{\frac{\pi}{2}} x \log(1-\cos x) \,dx $

"Nice" form/bounds for $\ln\prod_{i=1}^n \left(1 + \frac{x_i^2}{n}\right)$

How to calculate this integral $\int_{0}^{\infty} \left(\frac{x - 1}{\ln(x)}\right)^2 \cdot \frac{1}{1 + x^n} \,dx$

Is there a Closed-Form Solution for L2 Regularization Raised to a Power?

Integrating $\int_{0}^{\infty} \frac{\tanh \left(\frac{x}{2}\right)+\tanh (2 x)}{x}\left(e^{\frac{3 x}{2}}-1\right)^{2} e^{-4 x} \mathrm{~d}x$

How to calculate this sum $\sum_{n=1}^{\infty} \frac{H_n \cdot H_{n+1}}{(n+1)(n+2)}$

Integrating $\int_{0}^{1} \left(\frac{\arctan(x) - x}{x^2}\right)^2 \,dx$

How to integrate $\int_{0}^{\infty} \frac{1}{x^2 (\tan^2 x + \cot^2 x)} \,dx$

Closed form for $A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$

How to integrate $\int_{0}^{1} \int_{0}^{1} \int_{0}^{1}\frac{x^{4a - 1} \ln(x)}{\sqrt{yz} \cdot (1 + x^{2a}z + yzx^{4a} + yx^{2a})} \,dx \,dy\,dz$

How to integrate $\int_{0}^{1} \int_{0}^{1} \ln\left(\frac{1}{\sinh^2(x) + \cosh^2(y)}\right) \,dx\,dy$

How to integrate $\int_{0}^{1} \int_{0}^{\infty} \frac{\ln(1 + x^3) \cdot (\ln(1 + y) + \arctan(y))}{(1 + x^2)(1 + y)^2} \,dx\,dy$

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