It's a homework task and I can't get past the last step.
Task is to prove that $ B(x,y)=\int\limits_0^1 \frac{\tau^{x-1}+\tau^{y-1}}{(1+\tau)^{x+y}} \mathrm{d}\tau $
By substituting $t=\frac{1}{\tau+1}$ into beta integral $ B(x,y)=\int\limits_0^1 t^{x-1}(1-t)^{y-1}\mathrm{d}t $
and repeating it with substition $t=\frac{\tau}{\tau+1}$ and counting both results together it can be shown that $ 2B(x,y)=\int\limits_0^{+\infty} \frac{\tau^{x-1}+\tau^{y-1}}{(1+\tau)^{x+y}} \mathrm{d}\tau $
Can this result be transformed into the needed one? Or should the initial steps be different?