At the U.S. schools I have been at, a minor punishment for academic integrity such as this does not appear on your transcript and you are not obligated to mention it when you apply to other schools (unless they ask, which I don't expect that they will.) The same holds for most other "internal" disciplinary actions that the school takes.
This is particularly the case if the professor has not taken "formal" action. At many schools, there are two options for the professor. For minor penalties, the instructor can handle it personally with the student, as long as they can both come to an agreement. In this case, there is no permanent record of the penalty except with the professor.
For more serious penalties, or if the student and faculty don't agree, the instructor has to initiate a formal process. If the formal process results in a penalty, then additionally a disciplinary letter is put in your file (probably in the dean's office for your college). If there is another incident, the presence of the first letter will make the penalty worse. At my school, students are expelled after the third formal incident.
You can check with the professor, as jakebeal says, to find out whether a letter has been filed. As he says, you could tell the professor that you understand the penalty and accept it, and will not repeat it, but for your information you'd like to know if formal paperwork was filed. You could also ask this in person during office hours.