Asymptotic value of an integral using Mathematica

When I plug the integral
$ $ n \int_0^1 t^{r-1}(2t-t^2)(2-2t) dt$ $ , I get the following : $ $ 2^{(-1 + 2 n) }nt^{(-1 + r) }(Beta[n, n] – 4 Beta[1/2, 1 + n, n])$ $ where $ Beta(x,a,b)=\int_0^x u^{a-1}(1-u)^{b-1} du$ is the incomplete beta integral.I want to know whether Mathematica can give me intermediate steps and,more importantly,how I can get the asyptotic limit of the value of the integral as n approches infinity.Lots of thanks for any help or hints in advance