# integration – Closed form of beta distribution CDF when \$alpha + beta = 1\$

Is there a closed form of the regularized incomplete Beta function $$I_x(alpha, beta)$$ when $$alpha + beta = 1$$?

A nice closed form based on arcsin exists for the case where $$alpha + beta = 1/2$$ (see here). Is there a similar one for the more general case?

My motivation is that I’d like to find, given $$p in (0, 1)$$, the $$alpha$$ such that the $$Pr(X > 0.5) = p$$ where $$X sim Beta(alpha, 1- alpha)$$. I would like to do this efficiently for any $$p in (0, 1)$$ – which should be easy with the closed form of the regularized Beta function.