Testing Convergence of Integral How do I prove that the integral: $\int_0^1 x^p{} (1-x)^q dx$  converges or diverges?
Where do we have problems, and under what conditions they are vanished?
Hint :
Separate the integral into two integrals, one from $0$ to $1/2$ and the other from $1/2$ to $1$.
Then find for what values of $p$ and $q $ , both integrals converge.
HINT:

The integrand is the kernel of a  Beta$(p+1,q+1)$ density, and so the integral converges if $p&gt;-1, q&gt;-1$. 

In that case, the value is 
$$
  \int_0^1 x^p (1-x)^q dx
= B(p+1,q+1)
= \frac{\Gamma(p+1)\Gamma(q+1)}{\Gamma(p+q+2)},
$$
and in the case that $p$ and $q$ are integers, this is equal to $\frac{p!q!}{(p+q+1)!} = \frac{1}{\binom{p+q}{p}(p+q+1)}$