2kπ-π/2≤1/2x+π/3≤2kπ+π/2
2kπ-5π/6≤1/2x≤2kπ+π/6
4kπ-5π/3≤x≤4kπ+π/3
即
单调增区间为：【4kπ-5π/3,4kπ+π/3】,k∈Z