y = 2sin^2 x + cosx + 3
= 2(1 - cos^2 x ) + cosx + 3
= 2 - 2cos^2 x + cosx + 3
= - 2cos^2x + cosx + 5
= - 2(cos^2 x - 1/2 cosx ) + 5
= - 2[cos^2 x - 2 × 1/4 × cosx + (1/4)^2] + 1/8 + 5
= - 2( cosx - 1/4)^2 + 41/8
当 cosx = - 1/4 y有最大值,
ymax = -2(-1/4 + 1/4)^2 + 41/8 = 41/8
当 cosx = -1时,y有最小值,
ymin = -2×( -1 - 1/4)^2 + 41/8
= - 2×(- 5/4)^2 + 41/8
= - 25/8 + 41/8 = 2
所以,函数 y = 2sin^2 x + cosx + 3的值域是：y∈（2,41/8)
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