I’ve tried to do that but I’m still confused sorry
Inspection is the simplest way to do it.
But also you can just deduce b without any integration. You should be aware of that ex>0 for all x therefore 3ex+6e−2x>0 for all x.
If you're going to be integrating between 2 and b then you're finding the area under this curve between these two values. But since our funcion is always positive, the area between any two values is going to be positive (or in other words, non-zero). The only way the area is zero is if the length of the region over which we are integrating is zero. Hence b−2=0 and the result follows.
As pointed out by @ghostwalker, this workout seems odd (and rather trivial) so we're just wondering what the question was asking for in the first place.