Show that the random variable $U=\frac{X}{X+Y}$ has a uniform distribution on the range [0,1] when X and Y are independent random variables with the same exponential distribution.
I'm stuck at
$F_U\left(u\right)=P\left(U\le u\right)=P\left(\frac{X}{X+Y}\le u\right)$
Even if I assume that X=1 I get a divergent integral in the end
And If I assume that
$P\left(0\le \frac{X}{X+Y}\le 1\right)=P\left(0\le \:X\le \:X+Y\right)=P\left(-X\le Y\right)=1-P\left(Y\le X\right)$
Then I still get an divergent integral
So what should I do?