Your second paragraph pretty much resumes what Integrated Farming is. That is exactly what I do.

And everything you do in Organic Farming can be done in Integrated Farming - you only resort to chemical solutions if there are no other options left. And even then, only if you've gone beyond a certain point beyond which the damage will out-cost the benefits of the spraying.

Not that the other is not the case. Organic farming is not allowed to use chemical solutions even they are better for the environment.

edit: as others have pointed out, some chemicals are allowed. However the points remains, environmentally friendly means to control weeds is often not allowed in organic.

This is a common falacy. Organic forming can only use organic chemical solutions, some of which are far more toxic than their synthetic alternatives. Explore this idea by checking out strawberry production.

Berries iirc are particularly sensitive to various pesticides, more so than a lot of other produce. It's one of the few products I will seek out Organic over not, when I have them.

One can use chemical solutions in Organic Farming, if these are organic, e.g. rotenone, insect-killing viruses, bacteria and fungi.

In some rare cases, synthetic mineral solutions are allowed, like copper sulfate.