ClearAll;
g[x_]:= E^(-lam x);
L = Integrate[g[x],{x,0,b}];
f[x_]:= g[x]/L;
mu = Integrate[x f[x],{x,0,b}];
F[t_]:= Integrate[f[x],{x,0,t}];
Factor[L]
Factor[mu]
Solve[F[t]==1/2,t]
Factor[F[t]]