How can I evaluate the following integral? ($n \in R$, $n>0$)
$$\int_0^1x^{n-1}(1-x)^{n+1}dx$$
I was solving the following problem (as practice) in school:
Prove that the sum of $n+1$ terms of $$\frac{C_0}{n(n+1)} - \frac{C_1}{(n+1)(n+2)} + \frac{C_2}{(n+2)(n+3)}- \cdot\cdot\cdot = \int_0^1x^{n-1}(1-x)^{n+1}dx$$
Wolfram Alpha says that the integral evaluates to: $$\frac{\Gamma(n)\Gamma(n+2)}{\Gamma(2n+2)}=\int_0^1x^{n-1}(1-x)^{n+1}dx$$
To reiterate... :
1) How can I evaluate the indefinite integral of the above version?
2) How can I evaluate the definite integral?
3) How to prove the LHS - RHS equality in the aforementioned problem?