How to integrate $\int_0^1x^a(1-x)^bdx$ I have a question about an equation I am trying to integrate, the integral is:
$$\int_0^1 x^a (1 - x)^b ~dx,$$
where $a, b > 0$.
Any assistance with this problem would be appreciated.

The integral at hand is known as Euler's integral of the first kind. It's value, as function of $a$ and $b$ is know as beta function:
$$
 B(a+1,b+1) =   \int_0^1 x^a (1-x)^b \mathrm{d} x
$$
Integrating by parts, one can derive recurrence equations:
$$
    (a+1) B(a+1, b+1) = b B(a+2,b)
$$
Change of variables $x \mapsto 1-x$ gives $B(a,b) = B(b,a)$. Recurrence equations can be solved to give expression of the beta function in terms of ratio of Euler's $\Gamma$-functions:
$$
    B(a,b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}
$$
