I'm currently reading "Doing Bayesian data analysis" (2nd ed.) and he present a derivation on page 132 where he rewrites the beta function:
$$ B(a,b)p(z,N) = B(a+z, N-z+b) $$
Where
$$B(a,b) = \int_0^1 \theta^{a-1}(1-\theta)^{b-1} \; d\theta$$ and $$ p(z,N) = \int_0^1 \theta^{z}(1-\theta)^{N-z} \; d\theta $$
I can see that if I multiply the contents of both integrals directly, I can get the same result. But everywhere I look, says that most of the time $\int f(x) dx \int g(x) dx = \int f(x) g(x) dx$ does not hold.
So the questions is, what would be the right argument here for this being true? (If there is any explanation other than just solving the integrals for both versions and show that they are the same)