Categories
Tags
Closed-form
New post in Closed-form
A strange integral having to do with the sophomore's dream:
August 1st, 2022
definite-integrals
closed-form
calculus
real-analysis
integration
How to evaluate double limit of multifactorial $\lim\limits_{k\to\infty}\lim\limits_{n\to 0} \sqrt[n]{n\underbrace{!!!!\cdots!}_{k\,\text{times}}}$
September 24th, 2021
real-analysis
limits
factorial
closed-form
gamma-function
The closed form of $\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$
November 22nd, 2020
complex-analysis
definite-integrals
calculus
closed-form
real-analysis
Integral $\int_0^\infty\frac{\ln\left(1+x+\sqrt{x^2+2\,x}\right)\,\ln\left(1+\sqrt{x^2+2\,x+2}\right)}{x^2+2x+1}dx$
November 25th, 2020
closed-form
improper-integrals
logarithms
integration
calculus
Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$
August 1st, 2022
gamma-function
combinatorics
calculus
infinite-product
closed-form
Integral $\int_1^\infty\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)\frac{\mathrm dx}{\sqrt{x^2-1}}$
April 20th, 2022
hyperbolic-functions
conjectures
closed-form
trigonometry
integration
A difficult logarithmic integral ${\Large\int}_0^1\log(x)\,\log(2+x)\,\log(1+x)\,\log\left(1+x^{-1}\right)dx$
August 8th, 2020
integration
definite-integrals
logarithms
polylogarithm
closed-form
Closed form for $\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx$
August 1st, 2022
closed-form
improper-integrals
special-functions
Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $
June 6th, 2020
definite-integrals
calculus
integration
closed-form
Integral ${\large\int}_0^\infty\frac{\ln x}{1+x}\sqrt{\frac{x+\sqrt{1+x^2}}{1+x^2}}\ \mathrm dx$
August 1st, 2022
calculus
integration
definite-integrals
improper-integrals
closed-form
Closed Form for $~\int_0^1\frac{\text{arctanh }x}{\tan\left(\frac\pi2~x\right)}~dx$
June 9th, 2020
calculus
integration
definite-integrals
improper-integrals
closed-form
A conjectured closed form of $\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$
August 1st, 2022
calculus
integration
definite-integrals
improper-integrals
closed-form
Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$
August 1st, 2022
calculus
integration
logarithms
definite-integrals
closed-form
How can I prove $\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\frac{1}{n^2}\right)^{n^2}$?
September 15th, 2020
infinite-product
pi
closed-form
exponential-function
real-analysis
An integral involving Fresnel integrals $\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$
August 1st, 2022
closed-form
improper-integrals
special-functions
fresnel-integrals
Integral $\int_0^1\frac{1-x^2+\left(1+x^2\right)\ln x}{\left(x+x^2\right)\ln^3x}dx$
March 14th, 2020
calculus
integration
definite-integrals
improper-integrals
closed-form
Infinite Series $\sum\limits_{n=2}^{\infty}\frac{\zeta(n)}{k^n}$
August 1st, 2022
riemann-zeta
closed-form
complex-analysis
sequences-and-series
real-analysis
Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$
August 1st, 2022
analysis
integration
definite-integrals
zeta-functions
closed-form
Curious about an empirically found continued fraction for tanh
May 18th, 2020
hyperbolic-functions
continued-fractions
closed-form
number-theory
Integral $\int_0^\infty\frac{1}{x\,\sqrt{2}+\sqrt{2\,x^2+1}}\cdot\frac{\log x}{\sqrt{x^2+1}}\mathrm dx$
May 19th, 2021
integration
calculus
logarithms
improper-integrals
closed-form
Closed form for the sequence defined by $a_0=1$ and $a_{n+1} = a_n + a_n^{-1}$
September 22nd, 2020
recurrence-relations
closed-form
sequences-and-series
Evaluating $\int_0^{\frac\pi2}\frac{\ln{(\sin x)}\ \ln{(\cos x})}{\tan x}\ dx$
July 9th, 2020
calculus
integration
definite-integrals
improper-integrals
closed-form
Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$
August 1st, 2022
calculus
integration
improper-integrals
closed-form
harmonic-numbers
A closed-form expression for the integral $\int_0^\infty\text{Ci}^{3}(x) \, \mathrm dx$
March 18th, 2022
calculus
integration
definite-integrals
special-functions
closed-form
Conjectural closed-form representations of sums, products or integrals
August 1st, 2022
sequences-and-series
integration
soft-question
big-list
closed-form
A definite integral $\int_0^\infty\frac{2-\cos x}{\left(1+x^4\right)\,\left(5-4\cos x\right)}dx$
May 29th, 2020
calculus
integration
trigonometry
definite-integrals
closed-form
A closed form for $\int_0^\infty\frac{\ln(x+4)}{\sqrt{x\,(x+3)\,(x+4)}}dx$
July 21st, 2020
calculus
integration
definite-integrals
logarithms
closed-form
Integral $\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx$
August 1st, 2022
calculus
integration
definite-integrals
logarithms
closed-form
Evaluating $\int_0^\pi\arctan\bigl(\frac{\ln\sin x}{x}\bigr)\mathrm{d}x$
February 27th, 2020
calculus
integration
definite-integrals
logarithms
closed-form
How to find ${\large\int}_1^\infty\frac{1-x+\ln x}{x \left(1+x^2\right) \ln^2 x} \mathrm dx$?
July 21st, 2020
calculus
integration
definite-integrals
logarithms
closed-form
←
1
2
3
4
5
6
7
8
9
10
11
→