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Closed-form
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Improper Integral $\int_0^1\left(\left\{\frac1x\right\}-\frac12\right)\frac{\log(x)}xdx$
September 9th, 2020
closed-form
definite-integrals
sequences-and-series
integration
calculus
Is there a closed-form solution for $\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(3n+m)}$?
June 6th, 2021
closed-form
summation
definite-integrals
integration
Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $
August 1st, 2022
sequences-and-series
analysis
convergence-divergence
logarithms
closed-form
Find the value of $\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$
May 25th, 2020
calculus
integration
definite-integrals
improper-integrals
closed-form
How to prove $\int_0^1\tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$
October 26th, 2021
calculus
integration
definite-integrals
closed-form
A conjectural closed form for $\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!}$
August 1st, 2022
calculus
sequences-and-series
closed-form
conjectures
hypergeometric-function
What is the antiderivative of $e^{-x^2}$
August 1st, 2022
calculus
integration
indefinite-integrals
closed-form
error-function
Finite Sum $\sum\limits_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}$
August 1st, 2022
analysis
trigonometry
summation
closed-form
fractions
Integral $\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$
August 1st, 2022
calculus
integration
trigonometry
definite-integrals
closed-form
A closed form for $\int_0^1{_2F_1}\left(-\frac{1}{4},\frac{5}{4};\,1;\,\frac{x}{2}\right)^2dx$
August 1st, 2022
calculus
integration
special-functions
closed-form
hypergeometric-function
Prove $\int_0^{\infty }\frac1{\sqrt{x}}\left(\frac{\cos(\pi x^2)}{\sinh (\pi x)}-\frac1{\pi x}\right)dx=\frac{1}{\sqrt{2}}\zeta(\frac{1}{2})$
December 6th, 2020
integration
complex-analysis
definite-integrals
closed-form
riemann-zeta
Integral $\int_0^1\frac{\ln x}{\left(1+x\right)\left(1+x^{-\left(2+\sqrt3\right)}\right)}dx$
March 18th, 2020
definite-integrals
logarithms
closed-form
calculus
integration
Closed form for $\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm dx$
August 1st, 2022
calculus
closed-form
definite-integrals
logarithms
integration
Closed form for improper definite integral involving trig functions and exponentials?
August 23rd, 2020
definite-integrals
closed-form
Closed form for $\sum_{n=1}^\infty\frac{\psi(n+\frac{5}{4})}{(1+2n)(1+4n)^2}$
July 12th, 2020
sequences-and-series
special-functions
gamma-function
closed-form
combinatorics
Prove $\int\limits_{0}^{\pi/2}\frac{dx}{1+\sin^2{(\tan{x})}}=\frac{\pi}{2\sqrt{2}}\bigl(\frac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\bigr)$
August 1st, 2022
calculus
integration
definite-integrals
improper-integrals
closed-form
Is it possible to simplify $\frac{\Gamma\left(\frac{1}{10}\right)}{\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$?
August 1st, 2022
closed-form
special-functions
calculus
gamma-function
How can I prove this closed form for $\sum_{n=1}^\infty\frac{(4n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^n}$
August 1st, 2022
conjectures
closed-form
sequences-and-series
calculus
hypergeometric-function
What's the limit of $\sqrt{2 + \sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2 + ...}}}}}} $?
November 24th, 2020
calculus
real-analysis
limits
closed-form
nested-radicals
Computing $ \int_0^\infty \frac{\log x}{\exp x} \ dx $
May 5th, 2020
calculus
integration
closed-form
Evaluating $\int_0^\infty \frac{dx}{\sqrt{x}[x^2+(1+2\sqrt{2})x+1][1-x+x^2-x^3+...+x^{50}]}$
August 1st, 2022
calculus
integration
definite-integrals
improper-integrals
closed-form
Conjecture $\int_0^1\frac{dx}{\sqrt[3]x\,\sqrt[6]{1-x}\,\sqrt{1-x\left(\sqrt{6}\sqrt{12+7\sqrt3}-3\sqrt3-6\right)^2}}=\frac\pi9(3+\sqrt2\sqrt[4]{27})$
August 1st, 2022
calculus
integration
closed-form
conjectures
hypergeometric-function
Infinite Series $\sum\limits_{n=1}^\infty\frac{(H_n)^2}{n^3}$
August 1st, 2022
real-analysis
sequences-and-series
complex-analysis
closed-form
harmonic-numbers
Calculate the following infinite sum in a closed form $\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$?
August 1st, 2022
calculus
sequences-and-series
trigonometry
summation
closed-form
Prove $\int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2 $
August 13th, 2020
calculus
integration
definite-integrals
improper-integrals
closed-form
Integral of Combination Log and Inverse Trig Function
September 18th, 2020
definite-integrals
improper-integrals
calculus
closed-form
integration
Remarkable logarithmic integral $\int_0^1 \frac{\log^2 (1-x) \log^2 x \log^3(1+x)}{x}dx$
January 11th, 2022
integration
polylogarithm
zeta-functions
closed-form
sequences-and-series
Evaluating $\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx$
October 12th, 2020
definite-integrals
integration
real-analysis
closed-form
A closed form of $\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx$
October 10th, 2020
golden-ratio
closed-form
definite-integrals
integration
calculus
Do harmonic numbers have a “closed-form” expression?
August 1st, 2022
abstract-algebra
functions
closed-form
harmonic-numbers
number-theory
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