The special challenge confronting the undergraduate college is one of shaping an “integrated core” of common learning. It is the object clause of include.

The subject of this object clause is the special challenge, the predicate is is, and the predicate is one of shaping an “integrated core” of common learning. confronting modifies and restricts the subject the special challenge, so only participles can be used, and there is an active relationship between the two, so only ing participle can be used.