New posts in closed-form

Infinite Series $\sum\limits_{k=1}^{\infty}\frac{k^n}{k!}$

calculus real-analysis sequences-and-series summation closed-form

An integral involving Fresnel integrals $\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$

special-functions improper-integrals closed-form fresnel-integrals

Conjectural closed-form representations of sums, products or integrals

sequences-and-series integration soft-question big-list closed-form

Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$

calculus integration definite-integrals closed-form hypergeometric-function

Evaluate $ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}dx $

calculus integration complex-analysis definite-integrals closed-form

Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $

calculus integration definite-integrals closed-form polylogarithm

Integral $\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$

calculus integration trigonometry definite-integrals closed-form

Integral $\int_0^{\pi/2}\arctan^2\left(\frac{6\sin x}{3+\cos 2x}\right)\mathrm dx$

calculus integration trigonometry definite-integrals closed-form

Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$

calculus combinatorics gamma-function closed-form infinite-product

A conjectural closed form for $\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!}$

calculus sequences-and-series closed-form conjectures hypergeometric-function

How to find the integral $\int_{0}^{\infty}\exp(- (ax+b/x))\,dx$?

real-analysis calculus definite-integrals improper-integrals closed-form

How can I prove this closed form for $\sum_{n=1}^\infty\frac{(4n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^n}$

calculus sequences-and-series closed-form conjectures hypergeometric-function

Closed form for $\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm dx$

calculus integration logarithms definite-integrals closed-form

Closed form for $\sum_{k=0}^{l}\binom{k}{n}\binom{k}{m}$

discrete-mathematics summation binomial-coefficients closed-form

Prove $\int\limits_{0}^{\pi/2}\frac{dx}{1+\sin^2{(\tan{x})}}=\frac{\pi}{2\sqrt{2}}\bigl(\frac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\bigr)$

calculus integration definite-integrals improper-integrals closed-form

Integral $\int_0^\infty\frac{\operatorname{arccot}\left(\sqrt{x}-2\,\sqrt{x+1}\right)}{x+1}\mathrm dx$

calculus integration trigonometry improper-integrals closed-form

A conjectured closed form of $\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$

calculus integration definite-integrals improper-integrals closed-form

Integral $\int_{-1}^{1} \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} - sx + 1} \right) \, \mathrm dx$

calculus integration definite-integrals closed-form

Conjectured value of a harmonic sum $\sum_{n=1}^\infty\left(H_n-\,2H_{2n}+H_{4n}\right)^2$

calculus sequences-and-series closed-form conjectures harmonic-numbers

Infinite Series $\sum\limits_{n=1}^\infty\frac{(H_n)^2}{n^3}$

real-analysis sequences-and-series complex-analysis closed-form harmonic-numbers