calculus - Integration of $int_0^infty sqrt x e^{-x^2} dx$

I am having a really hard time trying to figure out what to do with this. I feel like I've tried everything but I'm obviously missing something. Any suggestions?

$$\int_0^\infty \sqrt x\ e^{-x^2}\ dx$$

These are all the techniques I have learned in class/can choose from: integration by parts, u-substitution, partial fractions, trig substitution. I don't know anything about the gamma function and am not allowed to use it.