New posts in beta-function

Evaluating $~\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~$ and $~\int_0^1\sqrt[n]{\frac{1+x^2}{1-x^2}}~dx$

calculus integration definite-integrals closed-form beta-function

Prove the following beta integral identity

integration gamma-function harmonic-analysis beta-function

Simplifying $\int_0^1 x^{a-1} (1-x)^{b-1} \, _2F_1\left(1,d;c+d+1;2-\frac{1}{x}\right) \, dx$

statistics definite-integrals hypergeometric-function beta-function

Sum: $\sum_{n=1}^\infty\prod_{k=1}^n\frac{k}{k+a}=\frac{1}{a-1}$

sequences-and-series algebra-precalculus closed-form beta-function

$ \int_0^\frac{\pi}{2}\ln^n\left(\tan(x)\right)\:dx$

integration definite-integrals gamma-function recursion beta-function

Baby rudin 8.21

analysis trigonometry gamma-function change-of-variable beta-function

How to Integrate $ \int^{\pi/2}_{0} x \ln(\cos x) \sqrt{\tan x}\,dx$

calculus integration definite-integrals gamma-function beta-function

Solving $\int_0^{\frac{\pi}{2}}\frac{1}{\sin^{2n}(x) + \cos^{2n}(x)}\:dx$

integration definite-integrals gamma-function trigonometric-integrals beta-function

Integrals related to $\int_0^{\pi} \left(\frac{\sin(\alpha u)^\alpha \sin((1-\alpha) u)^{1-\alpha}}{\sin u} \right)^{\rho/\alpha}du$

integration complex-analysis probability-distributions gamma-function beta-function

Incomplete hypergeometric function

special-functions hypergeometric-function beta-function

Harmonic number identity

calculus sequences-and-series binomial-coefficients harmonic-numbers beta-function

Methods to attack integrals that include $(1+x)^{a}\ln^{b}(1+x)$ in the integrand

integration definite-integrals gamma-function beta-function

Beta regression

regression maximum-likelihood beta-function

$ \frac{\Gamma(r)\Gamma(s)\Gamma(k)}{\Gamma(r+s+k)} $ as a nice integral?

gamma-function beta-function

Prove this integral $\int_0^\infty \frac{dx}{\sqrt{x^4+a^4}+\sqrt{x^4+b^4}}=\frac{\Gamma(1/4)^2 }{6 \sqrt{\pi}} \frac{a^3-b^3}{a^4-b^4}$

integration definite-integrals improper-integrals means beta-function

Proving the identity between Beta and Gamma functions using semi-group property of the Gamma.

probability gamma-function beta-function

Proving Renyi's result on the order statistics of the exponential distribution

probability exponential-distribution beta-function

Proving that $\int_0^\infty\Big(\sqrt[n]{1+x^n}-x\Big)dx~=~\frac12\cdot{-1/n\choose+1/n}^{-1}$

calculus integration definite-integrals closed-form beta-function

Prove: $\int_0^2 \frac{dx}{\sqrt{1+x^3}}=\frac{\Gamma\left(\frac{1}{6}\right)\Gamma\left(\frac{1}{3}\right)}{6\Gamma\left(\frac{1}{2}\right)}$

calculus integration definite-integrals elliptic-curves beta-function

Prove that $\int_0^1t^{p-1}(1-t)^{q-1}\,dt=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}$ for positive $p$ and $q$

integration gamma-function beta-function