Show that

$$\Gamma(\alpha) \Gamma(\beta)=\Gamma(\alpha+\beta) \int_{0}^{1} t^{\alpha-1}(1-t)^{\beta-1} d t, \quad \operatorname{Re} \alpha>0, \quad \operatorname{Re} \beta>0$$

where $\Gamma(z)$ denotes the Gamma function

$$\Gamma(z)=\int_{0}^{\infty} x^{z-1} e^{-x} d x, \quad \operatorname{Re} z>0$$