Definition of the gamma function for non-integer negative values The gamma function is defined as 
$$\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}dt$$
for $x>0$.
Through integration by parts, it can be shown that for $x>0$,
$$\Gamma(x)=\frac{1}{x}\Gamma(x+1).$$
Now, my textbook says we can use this definition to define $\Gamma(x)$ for non-integer negative values. I don't understand why. The latter definition was derived by assuming $x>0$. So shouldn't the whole definition not be valid for any $x$ value less than zero?
P.S. I have read other mathematical sources and most of them explain things in mathematical terms that are beyond my level. It would be appreciated if things could be kept in relatively simple terms.

The definition you gave is valid only for $x >0$, has you have pointed out. However, you can extend $\Gamma$ to negative non integer values by defining 
$$\Gamma(x) := \frac{1}{x}\Gamma(x+1) $$
whenever $x <0$, $x \notin \mathbb Z$. For example you get
$$\Gamma\left(-\frac{1}{2}\right) = -2 \Gamma\left(\frac{1}{2}\right) $$
and $\Gamma\left(\frac{1}{2}\right)$ is given by the integral you have presented.
