Mailing Books

I'm moving to NYC to be closer to my grandchildren. I'm getting rid of everything except things of very sentimental value and some difficult-to-replace teaching manuals. I plan to mail my adult children's childhood books and the teaching books. There are a LOT of books! It looks like it's cheapest to pack them in the biggest boxes possible up to 75 lbs. But what boxes are strong enough to survive the trip? I understand that the box may be inspected, so I should factor that in. I'd love advice!
And, oh my goodness, dealing with NYC co-op boards is every bit the nightmare it's chalked up to be! But that's another issue!