Integrate[ Cos[x]^50,{x, -1,1}]

Integrate[ Cos[x]^n,{x, -1,1}]

DiscretePlot[{Sqrt[2*Pi/n],0,(((Sqrt[π]) Gamma[((1+n)/2)])/(Gamma[1+(n/2)]))-((2 ((Cos[1])^(1+n)) Hypergeometric2F1[(1/2),((1+n)/2),((3+n)/2),((Cos[1])^2)])/(1+n))},{n,20,60}]