Newbetuts

A formula that counts exactly the twin prime averages occuring in an interval $[a,b]$ is surprisingly succinct!

On closed forms for the binomial sum $\sum_{n=1}^\infty \frac{z^n}{n^p\,\binom {2n}n}$ for general $p$?

How to prove this equality with closed form?

Conjecture $\sum_{n=1}^\infty\frac{\ln(n+2)}{n\,(n+1)}\,\stackrel{\color{gray}?}=\,{\large\int}_0^1\frac{x\,(\ln x-1)}{\ln(1-x)}\,dx$

Is there a known closed form solution to $\int_0^1\frac{\ln(1+x^{2n})}{1+x^2} \,dx$?

Generating function for cubes of Harmonic numbers

Integral $\int_0^1 \ln\left(\frac{1-x}{1+x}\right)\ln\left(\frac{1-x^2}{1+x^2}\right)\frac{dx}{x}$

How to prove this series about Fibonacci number: $\sum_{n=1}^{\infty }\frac{F_{n}}{2^{n}}=2$? [duplicate]

Infinite Series $\sum\limits_{n=1}^{\infty}\frac{1}{4^n\cos^2\frac{x}{2^n}}$

How to integrate $\int\limits_{0}^{\pi/2}\frac{dx}{\cos^3{x}+\sin^3{x}}$?

Integral $\int_0^1 \frac{(x^2+1)\ln(1+x)}{x^4-x^2+1}dx$

Integral $\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx$

Closed expression for $\int_{-\pi}^{\pi}\sqrt{a^{2}-2ab\cos(x)+b^{2}}dx $ [closed]

Integral $\int_{0}^{\infty }\!{\frac {t\ln \left( {t}^{2}+1 \right) }{{{\rm e}^ {4\,\pi\,t}}+1}}\,{\rm d}t$

Seeking closed form for infinite sum $\sum \limits_{ n=1 }^{ \infty }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! } }$

New posts in closed-form

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

Closed Form for $\int_0^1 \frac{\log(x)}{\sqrt{1-x^2}\sqrt{x^2+2+2\sqrt{2}}}dx$

Relations between definite integrals not having a known closed form

Close-form for integral $T(n)=\int_0^{\pi/2}\frac{1}{1+\sin^n(x)}dx$

Closed form $\int_{-1}^{1} \frac{\ln (\sqrt{3} x +2)}{\sqrt{1-x^{2}} (\sqrt{3} x + 2)^{n}}\ dx$

A formula that counts exactly the twin prime averages occuring in an interval $[a,b]$ is surprisingly succinct!

On closed forms for the binomial sum $\sum_{n=1}^\infty \frac{z^n}{n^p\,\binom {2n}n}$ for general $p$?

How to prove this equality with closed form?

Conjecture $\sum_{n=1}^\infty\frac{\ln(n+2)}{n\,(n+1)}\,\stackrel{\color{gray}?}=\,{\large\int}_0^1\frac{x\,(\ln x-1)}{\ln(1-x)}\,dx$

Is there a known closed form solution to $\int_0^1\frac{\ln(1+x^{2n})}{1+x^2} \,dx$?

Generating function for cubes of Harmonic numbers

Integral $\int_0^1 \ln\left(\frac{1-x}{1+x}\right)\ln\left(\frac{1-x^2}{1+x^2}\right)\frac{dx}{x}$

How to prove this series about Fibonacci number: $\sum_{n=1}^{\infty }\frac{F_{n}}{2^{n}}=2$? [duplicate]

Infinite Series $\sum\limits_{n=1}^{\infty}\frac{1}{4^n\cos^2\frac{x}{2^n}}$

How to integrate $\int\limits_{0}^{\pi/2}\frac{dx}{\cos^3{x}+\sin^3{x}}$?

Integral $\int_0^1 \frac{(x^2+1)\ln(1+x)}{x^4-x^2+1}dx$

Integral $\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx$

Closed expression for $\int_{-\pi}^{\pi}\sqrt{a^{2}-2ab\cos(x)+b^{2}}dx $ [closed]

Integral $\int_{0}^{\infty }\!{\frac {t\ln \left( {t}^{2}+1 \right) }{{{\rm e}^ {4\,\pi\,t}}+1}}\,{\rm d}t$

Seeking closed form for infinite sum $\sum \limits_{ n=1 }^{ \infty }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! } }$