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

Question

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.

Answer 1

Let $~t=x^{^\tfrac32}.~$ Then $\displaystyle\int_0^\infty\sqrt x~e^{-x^2}~dx=\frac23\int_0^\infty\exp\bigg(-t^{^\tfrac43}\bigg)~dt=\frac23\cdot\bigg(\frac34\bigg)!~$ This is based on

the fact that $n!=\mathcal G\bigg(\dfrac1n\bigg)$, where $\mathcal G(n)=\displaystyle\int_0^\infty~e^{-t^n}~dt.~$ See $\Gamma$ function for more details.

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I don't know anything about the gamma function and am not allowed to use it.

You can't not use it, since, as you can clearly see, there is no alternate way of expressing the result.