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Closed-form
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Integral $\int_0^\infty \frac{\ln(1+x+x^2)}{1+x^2}dx$
March 25th, 2022
integration
definite-integrals
closed-form
Closed form for ${\large\int}_0^\infty\frac{x-\sin x}{\left(e^x-1\right)x^2}\,dx$
August 1st, 2022
definite-integrals
closed-form
improper-integrals
integration
calculus
Calculating $\int_0^\infty \frac{\sin(x)}{x} \frac{\sin(x / 3)}{x / 3} \frac{\sin(x / 5)}{x / 5} \cdots \frac{\sin(x / 15)}{x / 15} \ dx$
August 1st, 2022
improper-integrals
products
trigonometric-integrals
integration
closed-form
Compute $\int_0^{1/2}\frac{\left(\operatorname{Li}_2(x)\right)^2}{x}dx$ or $\sum_{n=1}^\infty \frac{H_n^{(2)}}{n^32^n}$
October 23rd, 2021
harmonic-numbers
integration
sequences-and-series
polylogarithm
closed-form
Closed-form of an integral involving a Jacobi theta function, $ \int_0^{\infty} \frac{\theta_4^{n}\left(e^{-\pi x}\right)}{1+x^2} dx $
April 25th, 2020
integration
definite-integrals
closed-form
theta-functions
sequences-and-series
Harmonic Numbers series I
August 1st, 2022
harmonic-numbers
generating-functions
closed-form
catalan-numbers
sequences-and-series
Closed form for $n$th derivative of exponential of $f$
August 1st, 2022
derivatives
closed-form
partial-derivative
Evaluating $\int_0^{2\pi}\frac{dt}{\sqrt[4]{P(\cos t,\sin t)}}$
August 1st, 2022
definite-integrals
integration
calculus
closed-form
Evaluate $\int_0^1\frac{\ln(1-x)}{x}\text{Li}_3\left(\frac{1+x}{2}\right)dx$ , $\int_0^1\frac{\ln^2(1-x)}{x}\text{Li}_2\left(\frac{1+x}{2} \right)dx$
August 1st, 2022
integration
calculus
definite-integrals
improper-integrals
closed-form
Integral $\int_0^\infty \frac{\arctan(x^2)}{x^4+x^2+1}dx$
August 1st, 2022
closed-form
definite-integrals
integration
How to compute $\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$?
August 1st, 2022
integration
improper-integrals
definite-integrals
calculus
closed-form
Closed form for $\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}$ conjectured
August 1st, 2022
sequences-and-series
calculus
closed-form
summation
Evaluate $\int_{-\pi/4}^{\pi/4}\frac{x}{\sin x}\mathrm{d}x$
April 12th, 2021
definite-integrals
integration
sequences-and-series
closed-form
catalans-constant
Is there a closed-form equation for $n!$? If not, why not?
August 1st, 2022
factorial
combinatorics
algorithms
closed-form
Integral $\int_0^1\frac{x^{42}}{\sqrt{x^4-x^2+1}}\operatorname d \!x$
August 1st, 2022
integration
calculus
definite-integrals
closed-form
How to compute the monstrous $ \int_0^{\frac{e-1}{e}}{\frac{x(2-x)}{(1-x)}\frac{\log\left(\log\left(1+\frac{x^2}{2-2x}\right)\right)}{2-2x+x^2}dx} $
August 1st, 2022
closed-form
definite-integrals
integration
calculus
Solving the integral $\int_0^{\pi/2}\log\left(\frac{2+\sin2x}{2-\sin2x}\right)\mathrm dx$
August 1st, 2022
definite-integrals
closed-form
integration
Improper Integral $\int\limits_0^1\frac{\ln(x)}{x^2-1}\,dx$
October 23rd, 2020
integration
calculus
definite-integrals
improper-integrals
closed-form
How to integrate $\int_{0}^{\infty }{\frac{\sin x}{\cosh x+\cos x}\cdot \frac{{{x}^{n}}}{n!}\ \text{d}x} $?
August 1st, 2022
closed-form
integration
definite-integrals
improper-integrals
calculus
Closed form of $I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx$
August 1st, 2022
integration
calculus
closed-form
contour-integration
definite-integrals
Proving $\int_{\sqrt{\frac{3}{5}}}^1 \frac{\arctan (x)}{\sqrt{2 x^2-1} \left(3 x^2-1\right)} \, dx=\frac{3 \pi ^2}{160}$
August 1st, 2022
closed-form
special-functions
definite-integrals
integration
What is $\int_0^1 \frac{\log \left(1-x^2\right) \sin ^{-1}(x)^2}{x^2} \, dx$?
August 1st, 2022
closed-form
logarithms
integration
definite-integrals
What is the closed form of the $f$ with $f(1)=1$, $f(2)=7$ and $f(n)=7f(n-1)-12f(n-2)$ ($n\ge 3$)?
August 1st, 2022
recurrence-relations
recursion
closed-form
discrete-mathematics
Infinite Series $\sum\limits_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}$
August 1st, 2022
calculus
closed-form
real-analysis
sequences-and-series
complex-analysis
How to evaluate improper integral $\int_{0}^{\infty}\frac{\tan^{-1}{x}}{e^{ax}-1}dx$?
August 1st, 2022
definite-integrals
closed-form
improper-integrals
integration
What would qualify as a valid reason to believe there is a closed form?
August 1st, 2022
philosophy
closed-form
special-functions
soft-question
integration
Closed form solution to $\int_0^1\arctan^2(x)\,\sqrt{x}\,dx$
August 1st, 2022
definite-integrals
integration
calculus
trigonometry
closed-form
Integral $\int_0^\frac{\pi}{2} x^2\sqrt{\tan x}\,\mathrm dx$
August 1st, 2022
integration
definite-integrals
closed-form
alternative-proof
How to calclulate a derivate of a hypergeometric function w.r.t. one of its parameters?
August 1st, 2022
derivatives
hypergeometric-function
closed-form
special-functions
calculus
closed form of $\sum \frac{1}{z^3 - n^3}$
August 1st, 2022
closed-form
complex-analysis
sequences-and-series
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