Special functions

I have a relatively light question about special functions. As an example, it can be shown that ##\displaystyle \int_0^{\frac{\pi}{2}} \sqrt{\sin x} ~ dx = \frac{\sqrt{\pi} ~\Gamma (\frac{3}{4})}{2 \Gamma (\frac{5}{4})}##. Generally, the expression on the right would be taken as "the answer" to this problem. My question is, to what extent is this a complete answer? Isn't the gamma function technically just another integral that we don't know the value of? And if we derive the values of gamma numerically, why don't we just numerically evaluate the original integral to begin with?