you get the gamma function which is an integral in the PDF of the gamma distribution.
How do you integrate the gamma PDF from x=0 to x=1; when you have the gamma function also in the PDF to integrate
The gamma function taken at an integer value is just the factorial : Gamma(n)=(n-1)!
However, it's not the main problem. If you're looking for a general formula of the integral of this pdf between 0 and 1, you won't find it. It can only be expressed in terms of incomplete gamma function. I suggest you have a look at the wikipedia page about the gamma distribution !