Incomplete beta function

Remarks

Example: for $a = 2$ and $b = 1$, $B(x; 2, 1) = \int_0^x t\, dt = \frac{x^2}{2}$ and $B(2, 1) = \frac12$, so $I_x(2, 1) = x^2$.

Non-example of a valid argument: for $a = 0$ the integrand is $t^{-1}(1-t)^{b-1}$, whose integral diverges at $0$, so $B(x; 0, b)$ is not defined for $x > 0$.

The function $x \mapsto I_x(a, b)$ increases from $0$ to $1$ and is the distribution function of the beta distribution with parameters $a$ and $b$. The substitution $t \mapsto 1 - t$ gives the symmetry $I_x(a, b) = 1 - I_{1-x}(b, a)$. For positive integers $a$ and $n$ with $a \le n$, $I_p(a, n - a + 1)$ is the probability of at least $a$ successes in $n$ independent trials with success probability $p$.