Q:

Compute $E(\sqrt{X+Y})$ given that $X,Y$ are iid

Compute $E(\sqrt{X+Y})$ given that $X,Y$ are iid.
Assume $X,Y$ are iid both having an Exp($\lambda=1)$ distribution.
Although this is the only information provided in the question, I know that since they are independent, then the joint distribution must be $$f_{X,Y}(X,Y)=e^{-x} e^{-y}$$ 
Then we must integrate:
$$\int_{0}^{\infty}\int_{0}^{\infty}e^{-x}e^{-y}(x+y)^{1/2}dxdy=\int_{0}^{\infty}e^{-y}\int_{0}^{\infty}e^{-x}(x+y)^{1/2}dxdy$$
This integral seems complicated to evaluate. I tried it out and it seems like I'll have to do parts atleast twice. Wolfram says the answer is $1.32934$ and I have $1.33$ as one of the options to this question.
I want to know if there's an easier way to evaluate this. Do I even need the integral? Or perhaps is there some trick to get the answer with this integral?

A:

One way of getting the integral slightly faster is to see that everything is in terms of $x+y$ only.
So let $z=x+y$, which ranges from $0$ to $\infty$, and then given $z$ we have $x$ ranges from $0$ to $z$. The Jacobian for this is just $1$.
Then the double integral becomes
$$
\int_0^{\infty}\int_0^ze^{-z}z^{\frac{1}{2}}\,\mathrm{d}x\,\mathrm{d}z=\int_0^{\infty}e^{-z}z^{\frac{3}{2}}\,\mathrm{d}z=\Gamma\left(\frac{5}{2}\right)=\frac{3\sqrt{\pi}}{4}$$

