Yes investigate away and if any proprieties exist then the people need to vote him out or Congress needs to exercise their powers to remedy the situation. Until and unless proprieties are found and acted upon Obama is the President and CINC.
We know from his own admission that obama was born with dual citizenship( assuming that he was even born in Hawaii). This alone disqualifies obama from being eligible to be the President under Article 2 of the Constitution. obama has manage ( with a full time staff of attorneys)to block all lawsuits against him seeking access to birth records to use in court against him. There is no question that he is not the legal President, the only problem is getting access to the records to prove it. obama IS NOT nor EVER can be the legal President of the United States. He is not the legal Commander -In- Chief.