g[x_]:= x(2-x)^2;
L = Integrate[g[x],{x,0,2}];
f[x_]:= g[x]/L;
mu = Integrate[x f[x],{x,0,2}];
F[t_]:= Integrate[f[x],{x,0,t}];
L
Factor[D[f[x],x]]
mu
F[t]