X1[t_,b_]:=x[t]/.DSolve[{2*x''[t]*x[t]-x'[t]^2-1==0,x[-1]==b,x[1]==b},x[t],t][[1]]

X1[t,b]

Manipulate[Plot[((1-2 (b^2)+2 b (Sqrt[-1+(b^2)])-(t^2))/(2 (-b+(Sqrt[-1+(b^2)])))),{t,-5,5}],{b,1,2}]

X2[t_,b_]:=x[t]/.DSolve[{2*x''[t]*x[t]-x'[t]^2-1==0,x[-1]==b,x[1]==b},x[t],t][[2]]

X2[t,b]

Manipulate[Plot[((-1+2 (b^2)+2 b (Sqrt[-1+(b^2)])+(t^2))/(2 (b+(Sqrt[-1+(b^2)])))),{t,-5,5}],{b,1,2}]

L[t_,x_,y_]:=Sqrt[x*(1+y^2)]

L[t,X1[t,b],D[X1[t,b],t]]

fL1[b_]:=Integrate[L[t,X1[t,b],D[X1[t,b],t]],{t,-1,1},Assumptions->b>=1]

FullSimplify[fL1[b]]

fL2[b_]:=Integrate[L[t,X2[t,b],D[X2[t,b],t]],{t,-1,1},Assumptions->b>1]

fL2[b]

fL1[1]