(1)

\begin{align*} \int_{0}^{\frac{\pi}{2}}\sin^{x}t\cos^{y}tdt & =\frac{1}{2}\int_{0}^{\frac{\pi}{2}}s^{\frac{x-1}{2}}(1-s)^{\frac{y-1}{2}}ds\qquad,\qquad s=\sin^{2}t\\ & =\frac{1}{2}B\left(\frac{x+1}{2},\frac{y+1}{2}\right) \end{align*}

(2)

\begin{align*} \int_{0}^{\infty}\frac{t^{x}}{(1+t)^{y}}dt & =\int_{0}^{1}s^{y-x-2}(1-s)^{x}ds\qquad,\qquad s=\frac{1}{1+x}\\ & =B(x+1,y-x-1) \end{align*}

(3)

\begin{align*} \int_{-1}^{1}(1+t)^{x}(1-t)^{y}dt & =2^{x+y+1}\int_{0}^{1}s^{x}(1-s)^{y}ds\qquad,\qquad s=\frac{1+t}{2}\\ & =2^{x+y+1}B(x+1,y+1) \end{align*}