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You will need to file a non-resident return for the other state to report the income earned during your internship.  The non-resident state will only tax you on the internship income. 

You will also file a return with your resident state to report all income (including the internship).  Your resident state will allow a credit or adjustment for any tax paid to the non-resident state.  (You will not be taxed on the same income by both states.)