1. To observe or to inquire into details; to analyze according to a method: The police are investigating the robbery that took place in a jewelry store.

Sherlock Holmes investigated many criminal cases by following the footprints left at the scene of the crimes.

2. Etymologically, "following the footprints" is exactly what an investigator should do, because investigate comes from Latin in-, "in, into" + vestigium, "footprint, track, trace"; literally, "to follow footprints"; therefore, to search into something.