f[x_,y_]:=y^2+3*x^2-1

y0=1

x0=1

g1[x_]:=y0+Integrate[f[t,y0],{t,x0,x}]

FullSimplify[g1[x]]

g2[x_]:=y0+Integrate[f[t,g1[t]],{t,x0,x}]

FullSimplify[g2[x]]

g3[x_]:=y0+Integrate[f[t,g2[t]],{t,x0,x}]

FullSimplify[g3[x]]