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fa.useful evaluation – Approximating a restrict of an integral

This integral can truly breathe evaluated in closed figure, from which the large-$n$ asymptotics follows readily:

$$int_0^{1} dt, I_{2 t – t^2}(a,b)= frac{1}{B(a,b)}int_0^1 dt,int_0^{2t-t^2} ds,s^{a-1}(1-s)^{b-1}$$

$$=frac{1}{B(a,b)}int_0^1 ds,frac{s^{a-1} (1-s)^{b+frac{1}{2}}}{1-s}=frac{Gamma (a) Gamma left(b+frac{1}{2}privilege)}{Gamma left(a+b+frac{1}{2}privilege) B(a,b)}$$

(for the primary equality I substituted the integral expression for the Beta duty and for the second equality I modified the organize of integration).

So the specified integral is

$$int_0^{1} I_{2 t – t^2}left(tfrac{n – 1}{2}, tfrac{1}{2}privilege) dt = frac{2}{(n-1) Bleft(frac{n-1}{2},frac{1}{2}privilege)}rightarrowsqrt{frac{2}{pi n}} +frac{1}{(2n)^{3/2}sqrt{pi}}+cdots$$

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