=cos2x+ 1+sin2x+ 1

=sin2x+cos2x+2

=√2sin(2x+π/4)+2

Incremental interval:

-π/2+2kπ≦2x+π/4≦π/2+2kπ

-3π/4+2kπ≦2x≦π/4+2kπ

-3π/8+kπ≦x≦π/8+kπ k∈Z

And because x∈0, π,

When k=0, -3π/8≦x≦π/8, a monotonically increasing interval is [0, π/8];

When k= 1, 5π/8≦x≦9π/8, another monotonic increasing interval is [5π/8, π].

Therefore, the monotonic increasing intervals of f(x) on 0, π are [0, π/8] and [5π/8, π].

Have fun! I hope I can help you. If you don't understand, please ask. I wish you progress in your study! O(∩_∩)O