f(x)=sin⁴x+2sinxcosx-cos⁴x
=sin(2x)-(cos⁴x-sin⁴x)
=sin(2x)-(sin²x+cos²x)(cos²x-sin²x)
=sin(2x)-(cos²x-sin²x)
=sin(2x)-cos(2x)
=√2sin(2x-π/4)
2kπ-π/2≤2x-π/4≤2kπ+π/2 (k∈Z)時(shí),函數(shù)單調(diào)遞增,此時(shí)
kπ-π/8≤x≤kπ+3π/8 (k∈Z)
函數(shù)的單調(diào)遞增區(qū)間為[kπ-π/8,kπ+3π/8] (k∈Z).