$${B(z_1,z_2) = \frac{\Gamma(z_1)\Gamma(z_2)}{\Gamma(z_1 + z_2)}}$$
$${B(z_1,z_2) = \int_{0}^{1} t^{z_1 - 1} (1 - t)^{z_2 - 1} dt }$$
$${B(z_1,z_2) = 2 \int_{0}^{\frac{\pi}{2}} (\sin t)^{2z_1 - 1} (\cos t)^{2z_2 - 1} dt }$$
$${B(z_1,z_2) = n \int_{0}^{1} t^{n z_1 - 1} (1 - t^n)^{z_2 - 1} dt }$$
$${B(z_1,z_2) = \int_{0}^{\infty} \frac{t^{z_1 - 1}}{(1 + t)^{z_1 + z_2}} dt }$$
$${B(z,z) = \frac{1}{z} \int_{0}^{\frac{\pi}{2}} \frac{1}{ \left( \sqrt[z]{\sin t} + \sqrt[z]{\cos t} \right)^{2z} } \ dt }$$
$${\frac{1}{a} B\left(b-\frac{s}{a},\frac{s}{a}\right) = \int_{0}^{\infty} \frac{t^{s - 1}}{(1 + t^a)^{b}} dt }$$
$${\int_{0}^{\frac{\pi}{2}} (\cos t)^{x - 1} \cos yt \ dt = \frac{\pi}{2^x x B \left( \frac{x+y+1}{2} , \frac{x-y+1}{2} \right) } }$$
$${\int_{0}^{\pi} (\sin t)^{x - 1} \sin yt \ dt = \frac{\pi \sin \left( \frac{\pi y}{2} \right) }{2^{x-1} x B \left( \frac{x+y+1}{2} , \frac{x-y+1}{2} \right) } }$$
$${\int_{0}^{\pi} (\sin t)^{x - 1} \cos yt \ dt = \frac{\pi \cos \left( \frac{\pi y}{2} \right) }{2^{x-1} x B \left( \frac{x+y+1}{2} , \frac{x-y+1}{2} \right) } }$$
$${\int_{0}^{\pi} (\cos t)^{x - 1} \sin yt \ dt = \frac{\pi \cos \left( \frac{\pi y}{2} \right) }{2^{x-1} x B \left( \frac{x+y+1}{2} , \frac{x-y+1}{2} \right) } }$$
$${B(x,y) = \sum_{n=0}^{\infty} \frac{(1-x)_n}{(y+n) n!} }$$
\({(x)_n}\) is the rising factorial