(1)

\begin{align*} B\left(0;\alpha,\beta\right) & =\int_{0}^{0}t^{\alpha-1}\left(1-t\right)^{\beta-1}dt\\ & =0 \end{align*}

(2)

\begin{align*} B\left(1;\alpha,\beta\right) & =\int_{0}^{1}t^{\alpha-1}\left(1-t\right)^{\beta-1}dt\\ & =B\left(\alpha,\beta\right) \end{align*}

(3)

\begin{align*} B\left(z;\alpha,1\right) & =\int_{0}^{z}t^{\alpha-1}\left(1-t\right)^{1-1}dt\\ & =\int_{0}^{z}t^{\alpha-1}dt\\ & =\left[\frac{t^{\alpha}}{\alpha}\right]_{0}^{z}\\ & =\frac{z^{\alpha}}{\alpha} \end{align*}

(4)

\begin{align*} B\left(z;1,\beta\right) & =\int_{0}^{z}t^{1-1}\left(1-t\right)^{\beta-1}dt\\ & =-\int_{0}^{z}\left(1-t\right)^{\beta-1}dt\\ & =-\int_{1}^{1-z}t^{\beta-1}dt\\ & =-\left[\frac{t^{\beta}}{\beta}\right]_{1}^{1-z}\\ & =\frac{1}{\beta}\left(1-\left(1-z\right)^{\beta}\right) \end{align*}

(5)

\begin{align*} B\left(z;\frac{1}{2},0\right) & =\int_{0}^{z}t^{\frac{1}{2}-1}\left(1-t\right)^{0-1}dt\\ & =\int_{0}^{z}\frac{t^{-\frac{1}{2}}}{1-t}dt\\ & =\int_{0}^{z}\frac{t^{-\frac{1}{2}}}{1-t}dt\\ & =2\int_{0}^{\sqrt{z}}\frac{1}{1-t^{2}}dt\cmt{t^{\frac{1}{2}}\rightarrow t}\\ & =2\left[\tanh^{\bullet}\left(t\right)\right]_{0}^{\sqrt{z}}\\ & =2\tanh^{\bullet}\left(\sqrt{z}\right) \end{align*}

(6)

\begin{align*} B\left(z;\frac{1}{2},\frac{1}{2}\right) & =\int_{0}^{z}t^{\frac{1}{2}-1}\left(1-t\right)^{\frac{1}{2}-1}dt\\ & =\int_{0}^{z}\frac{1}{\sqrt{t}\sqrt{1-t}}dt\\ & =2\int_{0}^{\sqrt{z}}\frac{1}{\sqrt{1-t^{2}}}dt\cmt{t^{\frac{1}{2}}\rightarrow t}\\ & =2\left[\sin^{\bullet}\left(t\right)\right]_{0}^{\sqrt{z}}\\ & =2\sin^{\bullet}\left(\sqrt{z}\right) \end{align*}