I will be a bit more direct than the other answers.
Do you think this would violate an academic integrity policy?
Most likely, yes, this is considered cheating.
JeffE is correct that it depends on course policy. However, for most courses, looking up the answer to a problem and then paraphrasing it in your own words is not allowed. Especially without crediting the source. So check with your instructor, but unfortunately, you are already in the gray area of probably violating the rules.
I honestly retain and understand better from this
Are you sure this is true? It does sound like you are rationalizing. This depends a lot on the subject, and probably on the person, so we cannot judge for sure. However, it is very common to feel like one is learning by reading / paraphrasing the answers, and to not actually be learning; then you only find out a few weeks later when you are presented with an exam problem (or try to solve a problem without internet aid), and find you are stuck on the very basics.
It also depends on how much effort you put in before looking up the solution online. If you are articulating exactly where you are stuck and actively involved in the learning process, this can be effective. But if you are only passively involved (only trying to find the answer, rather than to grasp and discover for yourself the underlying concepts), then this is not effective.