Suppose $$f(x_{1},x_{2})=\frac{1}{\Gamma(\alpha)\Gamma{(\beta})\theta^{\alpha+\beta}}\exp(-\frac{x_1+x_2}{\theta^{\alpha+\beta}})$$ show that $y_{1}=\frac{x_1}{x_1+x_2}$ has the distribution of $\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}y_1^{\alpha -1}(1-y_1)^{\beta-1}$ for $0<y<1$. hint : $y_{2}=x_1+x_2$
I was doing it till I got stuck at the integral bellow $$\frac{y_1^{\alpha -1}(1-y_1)^{\beta-1}}{\Gamma(\alpha)\Gamma(\beta)}\int_{0}^{\infty} \frac {y_2^{\alpha+\beta-1}}{\theta^{\alpha+\beta}} \exp(-\frac{y_2}{\theta}) dy_2$$