Convergence of beta function

Question

Show that $$B(p, q) = \int_0^1 x^{p-1} (1-x)^{q-1} dx $$ converges for $p,q > 0$

I am not too sure how to do this, so any hints and solutions would be appreciated.

First of all, is it true to say that we have convergence for $p,q \geq 1$ since the integrand is continuous?

Now we just have to check for $p, q<1$.

I have tried to partition the integral, as we have discontinuous at $0$ if $p<1$ and at $1$ if $q<1$.

So I have done something that has been done in previous problems, to write $$B(p,q) = \int_0^{0.5} x^{p-1} (1-x)^{q-1} dx + \int_{0.5}^1 x^{p-1} (1-x)^{q-1} dx$$ but then I am not too sure what to do from here...

Answer 1

For \begin{align*} \int_{0}^{1/2}x^{p-1}(1-x)^{q-1}dx, \end{align*} the term $(1-x)^{q-1}$ is good on $[0,1/2]$, there is no denominator problem here, and $(1-x)^{q-1}\leq C$ for $x\in[0,1/2]$, $C>0$ is some constant.

So \begin{align*} \int_{0}^{1/2}x^{p-1}(1-x)^{q-1}dx\leq C\int_{0}^{1/2}x^{p-1}dx=\dfrac{C}{p}x^{p}\bigg|_{x=0}^{x=1/2}<\infty, \end{align*} for $0<p<1$.