to examine, study, or inquire into systematically; search or examine into the particulars of; examine in detail.

to search out and examine the particulars of in an attempt to learn the facts about something hidden, unique, or complex, esp. in an attempt to find a motive, cause, or culprit

The police are investigating the murder.

to make inquiry, examination, or investigation.

noun

agent , analyst , attorney , auditor , detective , examiner , gumshoe * , hound * , inquirer , inspector , plainclothes officer , police , private detective , private eye , prosecutor , researcher , reviewer , sherlock holmes , sleuth , snooper , spy , tester , undercover cop , inquisitor , prober , querier , quester , questioner