Beta distribution "support" over interval $[0, 1]$ .... why will it not integrate to $1$?

Question

I am trying to understand the Beta probability distribution. According to the textbook, the distribution has "support" over the interval $[0, 1]$. My understanding is that this is a Probability Density Function (PDF). If that is the case, then why when we compute the integral of this from 0 to 1 is the result not equal to 1? $$ \operatorname{Beta}(x\mid a,b) = \frac1{B(a,b)}x^{a-1}(1-x)^{b-1}$$ I have tried integrating it in a number of different ways. First, I worked out the integral. Then I used a calculus approximation technique (the Trapezoid Rule with the number of trapezoids equal to $200$). With both of these attempts, I got the same answer (which is no where near $1$): $0.009995$

Thanks in advance

Answer 1

By very definition $$\int_0^1 x^{a-1}(1-x)^{b-1}\, dx=:B(a,b).$$ Consequently by diving both sides by $B(a,b)$ one obtains $$\int_0^1 \frac{1}{B(a,b)}x^{a-1}(1-x)^{b-1}\, dx=1$$

Answer 2

Since you did this with $a=b=1/2,$ I suspect the vertical asymptotes at $0$ and $1$ (which occur when $a$ and $b$ are less than $1$) require more attention to detail than do the places where the density is more well behaved.