∵x^2+x+1=（x+1/2)^2+3/4≥3/4
∴3/4≤x^2+x+1≤1
∴ arcsin(3/4)≤arcsin(x^2+x+1)≤π/2
∴y=arcsin(x^2+x+1)的值域是[arcsin3/4,π/2]