i think putting tape over it is a good idea
Just glue the split parts together. If the books were on acid-free paper, Id recommend using PVA but cheap paperbacks can be fixed with carpenters glue.
It can be easier to just go to any dollar store and look for the tubes of clear contact paper. Theyre like lamination sheets, sticky on one side and smooth on the other. Doing a cut to size cover protector for the front and back is easy. You can do the inside front and back cover as well and trim around the edges. The roll is 3 feet long and only $1.
Tape would look sort of trashy and wont hold if it gets wet.